WEBVTT

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Exercise 3, and this one looks at proving theorem 3, also mentioned earlier.

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Now let me just quickly scroll over there to see where it was first brought up.

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Theorem 3, let's keep going somewhere in the middle, see what this one is.

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Example 2, 11, monotonic sequence, and there's theorem 3.

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So if limit as n approaches infinity of a n equals to l, and the function f

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is continuous at l, then what we have is, well limit as n approaches infinity of f of

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a n equals to f of l, and q where number is continuous.

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I'm going to prove this theorem in exercise 3, and I'll get to that in a bit and recall

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that I used it for this example where we have sine of a sequence, and we get as well, we

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know the limit of the sequence, we can just throw that inside, so we do the sine of the

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limit inside and solve it very quickly, like that.

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So let's just scroll back down and solve this, or prove that it is the case, alright, almost

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there, and yeah we're right here.

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So prove theorem 3, also mentioned earlier.

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So recall theorem 3, as I just went over, limit as n approaches infinity of a n equals

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l, and the function f is continuous at l, then we have this scenario, if the limit

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as n approaches infinity of f of a n equals to f of l, and here it's copied and

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based on my calculus book just for convenience.

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So what we know here as well, since we're given the limit, so since limit as n approaches

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infinity of a n equals to l, yeah since we're given this, then we must have, by the definition

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we must have a number, I'll write this a number, epsilon greater than zero, for actually I've

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written better, if for every epsilon, or a number epsilon greater than zero, there is an integer

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or a corresponding integer, as I again went over this definition many, many times, capital

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n such that, such that if, another if, if integer n is greater than this capital n integer, then

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we have the difference, then the difference a n minus the limit l is less than the number

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epsilon.

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Yeah, so since this is the case, since we were given the limit of a n as n approaches infinity

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is l, then we must have, for you again for any number epsilon greater than zero, there

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is an integer n such that, when you get larger than this integer, the difference a n minus

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l, the absolute value of this difference, is less than the number epsilon, so whatever

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it is, you can make it extremely small, again as I stated many, many times, if you make

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larger than that, then the difference gets smaller than that number that you picked, even

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if it's really, really small, again because the limit by definition exists.

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And also, so also since we were given that, also since f is continuous, is f is continuous

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at l, yeah so this at x equals l, or at l, so it's continuous at l, then what we

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must have, and all right here then, by definition, by again another precise

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definition of limits, or here I wrote it a bit better, I just typed it out, so and

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since f is continuous at l, by the precise definition of limits shown in my

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earlier videos, we must have the case right here, the limit as x approaches

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l of f of x must equal to, well f of l, so the limit exists in the function f of l

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equals to, or I mean the limit as x approaches l of f of x equals to f of l,

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if here, if for again similar to the above definition for sequences, if for every

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number epsilon greater than zero, there is now, instead of an integer, there is a

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number, we'll call this delta, so again number versus integer, number delta such that,

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I'll show you separately, this is a number delta greater than zero, it's also greater

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than zero, such that, yeah such that if we have f, the absolute value of the

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difference x minus l is going to be, yeah if it's less than this specific number

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delta, whatever it is greater than zero, then again instead of going just larger

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than capital N, because we're going to a specific point as opposed to going to

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infinity for these limits, we're going to go, yeah we're moving to l, so we're

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just looking at interval at l, so very very small, then the scenario we

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have is, then f of x minus f of l, this difference is going to be less than this

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epsilon number, yes that was a definition, but what I'm going to do is I'm going to

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just correlate these better with this above, so what I'm going to do is I'm going to

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replace, just switch these around, use the switch terminal, I'm going to call this

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delta, call this epsilon, so that this is going to be delta, and this is

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epsilon, so that these line up, notice how this is going to be exactly the same

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thing if we let an equal to x, yeah so thus what we have is, well that's the

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definition of continuous, so thus let's just say suppose now, so suppose we had

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n is greater than capital N, so let's say suppose n is greater than capital

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N, then what we have is, yeah then what we have is this scenario is the

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case, so then we have is, yeah absolute value of an minus l is less than epsilon,

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yeah so that's, this is what we have right here, but instead of just, just

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regurgitating this limit right here, what we could do is, we'll replace this x, so

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if this x is just an, what we could do is, well then the, then this is the case

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and, all right over here actually, yeah just right and, but what I'll do is

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I'll write this, but as instead of x I'll put an, and then what we have is

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f of an minus f of l is less than epsilon, yeah so this way right here is, instead of

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actually have this one and then it's everything is everything of this

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definition aligns up, so thus what we have is thus, so suppose this is the case

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also right then, well actually thus is better, thus we're replacing x with this

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so thus limit as an approaches l of f of an is equal to f of l by definition, I'll

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just write that here by definition of the above continuous function, so by

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definition or by precise definition, but notice this is the same thing, so a limit

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as an approaches l is the same thing as, this is same as limit, I'll just write this

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down, limit an approaches l of f of an, it just equals to limit as an approaches

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infinity of f of an, so as an approaches infinity it'd be exactly the same thing, so

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again this will just equal over there, so equals to f of l, so yes, I'll just

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circle this, thus hopefully you got your head around that, it's just, it's pretty

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straightforward one to understand these precise definition terminology, but yeah

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it's just pretty abstract and precise definition has always been a bit of a

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brain teaser, the way that they use these definitions, again it's all it has to do

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is with when you, just any number you pick there's always going to be a number

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such that this difference is always going to be less than that number and so on, so

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yes so this is the same thing, so in essence all we're doing here is well

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right here we just replace this this l with limit as an approaches infinity of

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an, and essentially just throw that in there, and then x will be the an, so we

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get that, so it's pretty straightforward, I'll worked around the long ways to get to

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that, same thing that we can intuitively look at, so there it is