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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:50:38
I guessed it that way since all intervals are closed on left
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:51:39
Now Suppose we are taking union of At's
0
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:52:29
The left end point will move from 2 and will get closer and closer to 1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:52:37
Right
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:52:47
And the right point will go from 5 closer to 3
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:52:51
But will never touch 1
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:53:02
As 1/t never becomes 0
0
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:53:08
Right
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:54:15
Right end point starts at 5 for A1,but become smaller later .So the largest set in union have right end point =1,So union has right end point=5.
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:54:36
As you drawed here
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:54:58
So far are you with me?
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:55:46
Can you explain this last thing?
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:55:55
The 2nd sentence
1
0
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:56:14
So A1 is (2,5)
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:56:18
Yes
1
0
869,532,694,688,706,600
EpsilonNought
8684
EpsilonNought
false
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:56:41
As t gets larger right end point will move from 5 to 3
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:57:06
Right but not touching
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0
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:57:12
Yes, exactly.
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1
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:57:20
But here we are taking the union
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:57:58
So on taking union ,it will contain A1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:58:13
And no other At have right end point larger than A1
0
1
869,532,694,688,706,600
EpsilonNought
8684
EpsilonNought
false
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:58:33
Right
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:58:43
So right end point of union must be A1's=5
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 14:59:14
So 5 closed interval right?
1
0
896,755,250,092,998,800
b3s4d
9845
b3s4d
false
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:59:21
Yes, Exactly
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:59:30
So union is (1,5>
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1
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 14:59:48
But we have to Show this is the case as well ,set theoretically.
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:00:12
And how do I do that
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:00:40
So first show (1,5> is a subset of the union
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:00:53
Pick a point x in (1,5>
0
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:01:16
Then 1<x<5
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:01:41
We can find a t such that (1+1/t)<x
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:02:20
So x is in At
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:02:51
If x is like 1.0000000001 then there is a number t that is smaller than x?
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:02:55
At I mean
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:03:17
No,we know x>1
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:03:18
t of course will get bigger but At will be smaller than x
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:03:46
Oh I meant 1 not 0
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0
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bbebe736909b460ba7d3fedaddca7b23
807,287,208,314,798,100
sreedev
01/21/2022 15:03:50
Yes,That's true.Let me think.
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:07:33
Do I have to prove it though? You said about assuming a number x inside an interval (1,5]
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0
896,755,250,092,998,800
b3s4d
9845
b3s4d
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:07:41
Or I can show it with that drawing?
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bbebe736909b460ba7d3fedaddca7b23
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Ratatouille
01/21/2022 15:09:48
why is (1,5] under consideration when 5 is not part of the union?
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:10:21
But it is part of the union
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:10:31
it's not
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:11:00
A1 is {2,5}
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:11:12
what's that? that's a set, not an interval
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:11:37
the interval is [2,5)
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:12:29
Sorry but that doesn't make sense to me, can you explain it to me?
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:12:44
ok when we put t=1, we get <2,5) right?
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:13:30
where by "<" i don't mean the less than sign, i mean that arrow bracket thing on the left
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:13:56
Why is 2 included but 5 not included in the interval
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:14:10
look at the definition of <a,b)
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:14:25
it's {x | a≤x<b }
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:14:31
emphasis on the <b
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:14:51
so when we have <2,5) we're only looking at numbers **strictly** less than 5
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:16:42
Are you sure?
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:17:05
yea
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:17:13
That's an interval and for A1 we can have the value 5
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:18:12
Sorry I'm getting really confused here
1
0
896,755,250,092,998,800
b3s4d
9845
b3s4d
false
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:18:38
it's alright
0
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:18:44
look at this definition
https://cdn.discordapp.c…6653/unknown.png
unknown.png
0
1
null
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205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:18:58
we're considering <2,5) which means a=2 and b=5, is that okay?
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:19:20
There is another definition too though
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:19:29
oh is there
0
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:19:37
https://cdn.discordapp.c…3726/unknown.png
unknown.png
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:20:01
right, but that one doesn't apply to A_t
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:20:02
Yea they are separated by a comma
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:20:07
look at this
https://cdn.discordapp.c…6069/unknown.png
unknown.png
0
1
null
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null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:20:14
Wait
1
0
null
null
null
null
null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:20:17
I am blind
1
0
null
null
null
null
null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:20:25
I didn't see the parenthesis
1
0
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:20:31
You are correct
1
0
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205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:20:32
yep haha
0
1
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null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:20:42
So the intersection will be (1;5)
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:20:59
no
0
1
null
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null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:21:28
for an element to be in the intersection
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:21:33
it has to be in each of the intervals
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help-1
bbebe736909b460ba7d3fedaddca7b23
543,222,734,021,197,800
Arsen
01/21/2022 15:21:34
nope
0
1
null
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null
null
null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:22:00
Wait
1
0
null
null
null
null
null
205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:22:06
I confused it with union
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:22:15
I meant union is (1;5)
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help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:22:34
that's right
0
1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:22:44
What about the intersection
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205,551
help-1
bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:23:02
should be [2,3]
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:23:44
I understand how we get 2
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:23:48
What about 3
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:24:14
two parts to this: prove that every element in [2,3] is in all of the intervals
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:24:31
secondly, show that every element *outside* [2,3] doesn't lie in at least one of the intervals
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:24:58
so first off, do you see that 3 is in all the intervals?
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:25:20
You are right
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0
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:25:33
If we take limit of 3+2/t we never touch 3
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:25:40
mhm
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1
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:25:50
I think I understand it
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0
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205,551
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:26:10
Is it okay if you explain me something small in the next one because I don't understand what's our interval?
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:26:17
sure
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:26:25
https://cdn.discordapp.c…3304/unknown.png
unknown.png
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:26:35
What's tN-{0.t} supposed to be
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bbebe736909b460ba7d3fedaddca7b23
832,714,326,548,021,400
Ratatouille
01/21/2022 15:26:58
that just means {2,3,4, ... }
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bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:27:19
No I know that one
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help-1
bbebe736909b460ba7d3fedaddca7b23
486,622,768,511,844,350
SmoothChick
01/21/2022 15:27:35
https://cdn.discordapp.c…6089/unknown.png
unknown.png
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