00:01
Okay, so we start with this theorem that every bounded monotonic sequence is convergent.
00:07
And we want to start by just saying what that means.
00:10
Well, so sort of just by pulling apart the definitions here, what we can say that this means that if a sequence, call it a -n, is either increasing or decreasing.
00:36
And so this is the monotonic part, either increasing or decreasing.
00:43
I need to say some aboundedness, and there are some constants, call them little m and big m, such that our little m is less than or equal to a .n, and that's less than equal to our big m for all n.
01:07
So meaning that each a .n lies, you know, in between this little m and this big m.
01:16
Then the conclusion is that then our sequence a .n converges.
01:22
So just to say a little bit more about what this means, well, maybe i'll draw a picture.
01:27
So let's just throw these on a number line.
01:29
Think about like our sequence being, almost like being a function that's taking values at these integer points.
01:43
So maybe our big m is a big m is a appear and then maybe we've got some we can just throw a little m anywhere down here.
01:49
And then maybe we can draw some sequence.
01:53
We want it to be, let's just say, increasing.
01:56
And then the idea is that, well, it's going to get maybe closer and closer to this big m, but it's never going to cross that.
02:07
So it's increasing, but it never, maybe it gets closer and closer this big m, but all of our a .n values stay below that big m.
02:18
Well, the conclusion then is that the sequence has to converge.
02:23
So i'm just going to write this in one more slightly different way.
02:27
So again, what does this theorem mean? it's just saying that if a sequence keeps increasing or decreasing but doesn't grow arbiterate.
02:51
Large, just another way of saying that it's bounded, then it must converge.
03:05
So that's basically saying what this theorem means in a few different ways.
03:10
Okay, so now we want to look at examples.
03:14
So part b asks you to give an example of a monotonic sequence that's bounded and to show that that converges.
03:26
So let's give an example.
03:28
So let a n equal n plus 1 for n greater than equal to 0.
03:37
So let's see.
03:38
So we want to, let's first try to show that this is bounded.
03:42
Well, that's pretty easy.
03:43
So we can say that since 1 is less than equal to n plus 1 for all n greater than equal to 0, which that's obvious, right? because n is at least, well, n is greater than equal to 0.
03:58
So we're just adding on something to one.
04:02
So that's going to make that at least that same size.
04:05
Well, then we can take the reciprocals.
04:09
Then 1 is greater than equal to 1 over n plus 1 for all, and greater than equal to 0.
04:17
What is this exactly saying? this is saying that 1 is greater than equal to a -n for all n.
04:25
So that tells us that this is our, now we're thinking of m, sorry, one is our little m.
04:33
So then, so this, a, n, our sequence a .n is bounded above by one.
04:45
Okay.
04:46
And then let's, we want to, we also want to lower bound...