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507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:26:44 | Please 🙏 | 1 | 0 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:29:10 | Hello? | 1 | 0 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:30:46 | I want to make g have no indices | 1 | 0 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 293,744,814,472,822,800 | Bunnies_ | 02/05/2022 08:31:14 | you'll want to square both sides | 0 | 1 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 08:31:14 | no | 0 | 1 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:31:44 | oh yes | 1 | 0 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 08:32:29 | $$\left(g ^\frac{1}{2}\right)^\frac{-1}{2}=g^\frac{-1}{4}$$ | 0 | 1 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 510,789,298,321,096,700 | TeXit | 02/05/2022 08:32:51 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 08:32:57 | You apply the same exponents to both sides | 0 | 1 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 08:33:10 | so one of your first ones just doesnt work out cus of this | 0 | 1 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:33:22 | Oh ic | 1 | 0 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:33:24 | Thanks | 1 | 0 | null | null | null | null | null | |||||
507 | help-10 | e4fa1722f9124db4833f519852c0101b | 886,259,475,911,180,300 | uuuu | 02/05/2022 08:33:28 | .close | 1 | 0 | 293,744,814,472,822,800 | Bunnies_ | 1349 | Bunnies_ | false | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:19:47 | is the inverse of y=f(x) x=f(y) | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:20:50 | no | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 521,326,998,195,273,700 | shreksdad | 02/05/2022 09:21:29 | y = f(x) makes no sense, they serve similar purpose | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:21:38 | hmm | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:21:43 | alright | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:22:26 | wait | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 521,326,998,195,273,700 | shreksdad | 02/05/2022 09:22:27 | f(x) means function | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 675,340,001,483,030,500 | tam | 02/05/2022 09:22:30 | @rechonchota its x=g(y) | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:22:39 | yes thats why i deleted it cause ik that | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 521,326,998,195,273,700 | shreksdad | 02/05/2022 09:22:44 | Ah ok | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:22:47 | srry i am learning about inverse functions | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:22:53 | and to make an inverse function | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:22:56 | the inverse of `f` if `f^(-1)`
where `f(x) = y` and `(f^(-1))(y) = x` | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:23:17 | you rearange to find x | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:23:22 | then you change x to y | 1 | 0 | 835,603,021,425,148,000 | rechonchota | 2320 | rechonchota | false | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:23:26 | and `f^(-1)` is just a notation for another function (like `g(y)`) | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:23:31 | ahh yes | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:23:47 | why do we change all x to make y | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:24:03 | like the inverse of y= 3/(x-1) | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:24:22 | is x= (3+y)/y | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:24:28 | but you chnage the y to x | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:25:20 | `3/(x-1) = y` means `x = (y + 3)/y` | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:25:39 | and `x = g(y) = (y+3)/y` is the inverse function | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:02 | oh | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:06 | but you can change y to x | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:26:08 | but since you usually use `x` as an argument and `y` as a function's value | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:10 | and say it is f-1 | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:26:11 | you just rename it | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 586,955,478,529,998,800 | rept1d | 02/05/2022 09:26:17 | so it is `y = g(x) = (x+3)/x` | 0 | 1 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:28 | perfect | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:34 | idky im getting so confused | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:38 | this has cleared it up | 1 | 0 | null | null | null | null | null | |||||
508 | help-10 | e76c0d60eed549b6884fca045b5cab44 | 835,603,021,425,148,000 | rechonchota | 02/05/2022 09:26:41 | thanks buddy | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:36:08 | How valid is this proof? "Suppose there exists $N_0$ such that $s_n \le t_n \forall n > N_0$. Prove that if lim$t_n$ and lim$s_n$ exists, then lim$s_n$ $\le$ lim$t_n$."
Proof:
\\\\
Let lim$t_n$ = t and lim$s_n$ = s. Let $\epsilon > 0$, and observe that $|t_n - t| < \epsilon$ $\forall n > N_0$, $|s_n - s| < \epsilon$ $... | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 510,789,298,321,096,700 | TeXit | 02/05/2022 10:38:50 | **dulg** | 197712244354711552.png | 0 | 1 | null | null | null | null | null | ||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:39:57 | \lim f(x) | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:40:00 | looks a lot better in latex. | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:40:20 | i will keep that in mind, thanks | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:41:15 | unknown.png | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:41:24 | How does this follow? | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:41:51 | you basically add - s -t to both sides | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:41:58 | oh nvm | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:42:02 | i was reading -s+t in head | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:42:38 | 0 | 1 | null | null | null | null | null | ||||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:42:49 | How does this follow from the above? | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:43:40 | If s is greater than s_n or t greater than t_n | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:43:44 | then ? | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:45:47 | ===
Something helpful for this proof might be to draw a rough diagram | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 10:45:54 | And to mark epsilon | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:57:07 | Good point. I kind of assume it since the textbook already went through it. Because of how absolute value is defined, where $|a| = a$ if a > 0, and $|a| = -a$ if a < 0, it follows that $-|a| \le a \le |a|$, since either $a = |a|$ or $a = -|a|$.
Thus: $-|s_n - s| \le s_n - s \le |s_n - s|$ $\Rightarrow$ $-|s_n - s| -t ... | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 510,789,298,321,096,700 | TeXit | 02/05/2022 10:57:12 | **dulg** | 197712244354711552.png | 0 | 1 | null | null | null | null | null | ||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:57:13 | .
@Shuri2060 ok wait | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:57:19 | i am a bit stumped | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:57:33 | realized something is a bit off halfway thru my last latex post | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 10:58:37 | wait for me while i post this idea on how to prove $|s_n - s| - t \le |t_n - t| -s$ | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 510,789,298,321,096,700 | TeXit | 02/05/2022 10:58:40 | **dulg** | 197712244354711552.png | 0 | 1 | null | null | null | null | null | ||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:04:30 | Consider case 1, where $s_n - s -t = |s_n - s| - t$. $s_n \le t_n$ implies $|s_n - s| - t \le t_n - t - s \le |t_n - t| -s$.
\\\\
Case 2, where I got stuck because I am not sure :(. Following the same process leads us to $s_n - s - t = -|s_n - s| - t \le t_n - t - s \le |t_n - t| - s$ | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 510,789,298,321,096,700 | TeXit | 02/05/2022 11:04:33 | **dulg** | 197712244354711552.png | 0 | 1 | null | null | null | null | null | ||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:05:26 | The ambiguity comes from where $|s_n - s| - t$ would fit in Case 2 | 1 | 0 | 321,445,119,171,756,000 | Shuri2060 | 9525 | a μ-er | false | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 510,789,298,321,096,700 | TeXit | 02/05/2022 11:05:28 | **dulg** | 197712244354711552.png | 0 | 1 | null | null | null | null | null | ||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:06:33 | I would say to restart your proof | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:06:43 | Because there really is no need to split cases | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:06:49 | What you need to do | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:06:52 | is draw a rough diagram | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:07:03 | and indicate what you want to show with epsilon pictorially | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:07:15 | what would that look like? | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:07:33 | I have never imagined real analysis using pictures | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:07:47 | hmmm | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:08:38 | the entire proof? or this absolute value thing? | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:09:08 | unknown.png | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:09:19 | Maybe all, cus there wasn't that much progress | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:09:37 | I'm just saying restart in terms of ideas | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:09:42 | Stuff will be reused | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:11:05 | In particular, like you said | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:11:18 | unknown.png | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:11:20 | We will take this | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:11:39 | Using this, we need to prove s <= t | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:12:01 | idk if the picture will help you see what you need to do with these 2 inequalities | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:13:42 | not sure about the picture | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 197,712,244,354,711,550 | dulg | 02/05/2022 11:14:05 | i suppose they are converging to a set number, and one is def greater than the other | 1 | 0 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:14:35 | Try to draw your own | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:14:45 | and mark an epsilon, basically | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:15:21 | @dulg ah actually, I see an error in your inital proof | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:15:45 | $$(\forall n > N_0)(\abs{s_n - s} < \varepsilon)$$
$$(\forall n > N_0)(\abs{t_n - t} < \varepsilon)$$ | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 510,789,298,321,096,700 | TeXit | 02/05/2022 11:15:48 | **Shuri2060** | 321445119171756033.png | 0 | 1 | null | null | null | null | null | ||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:15:54 | You make these claims. | 0 | 1 | null | null | null | null | null | |||||
509 | help-10 | 209372b213bd479ca6cc76c57845c087 | 321,445,119,171,756,000 | Shuri2060 | 02/05/2022 11:16:01 | They are not necessarily true --- think carefully why. | 0 | 1 | null | null | null | null | null |
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