1
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Finding limits is hard.
I wish I could just prove that they exist.

2
00:00:04,627 --> 00:00:10,380
[MUSIC].

3
00:00:10,380 --> 00:00:15,100
Here's a theorem that guarantees a
sequence converges.

4
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If the sequence is bounded and monotone,
then the limit exists.

5
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Why is this important?

6
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Well, I often can't tell whether a
sequence converges.

7
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But I may be able to show that a sequence
is both bounded and monotone in

8
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that I know it has a limit.

9
00:00:39,260 --> 00:00:43,090
But why should I believe that the monotone
conversions theorem is true?

10
00:00:43,090 --> 00:00:45,010
Well let's think about this geometrically.

11
00:00:45,010 --> 00:00:49,454
Suppose I've got a number line and I've
got terms of

12
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my sequence, x of 1, x of 2, x of 3.

13
00:00:52,820 --> 00:00:54,970
Let's pretend they're increasing.

14
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And let's pretend that the sequence is
bounded.

15
00:00:58,320 --> 00:01:00,652
So I know that the sequence

16
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never exceeds this value b.

17
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So I've got a sequence which is increasing
and bounded above.

18
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Well what can happen, right?

19
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As I go out further and further in that
sequence, I can

20
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never pass b and yet I have to keep moving
to the right.

21
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So hopefully it seems plausible that with
these conditions, this sequence can't

22
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help but converge to some limiting value
l.

23
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So hopefully it seems plausible but we
don't

24
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yet have the tools to give a formal proof.

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[SOUND]

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[SOUND]