00:01
So we have a sequence here, xn, and we're saying that it verkonjes to x, so i'm going to write that as like that, because it's like the opposite of convergence in a sense.
00:14
Verkonjes to x if there exists some positive epsilon such that for all natural n, whenever n is greater than or equal to n, it implies that xn minus x is less than epsilon.
00:45
So in natural language here, we're saying that for any radius of x, or excuse me, there is some radius of x such that all of the xn are within that radius.
01:07
And to put that more precisely, we're going to have our x here, and let's see, it says there exists some epsilon, so i'm going to draw an arbitrary epsilon radius of x.
01:24
This guy has length epsilon, this guy has length epsilon.
01:28
We're saying for all natural big n, whenever little n is bigger than big n, the distance between xn and x is less than epsilon.
01:42
In particular, we can pick the value 1, so if n is greater than or equal to 1, then xn is within epsilon of x, which really just means that all of the xn in general are within our range epsilon.
02:02
If your convention is that 0 is a natural number, as frankly it should be, really this collapses into there exists some epsilon greater than 0 such that xn is within epsilon of x for all values n.
02:27
Excuse me, that's an alarm.
02:31
And with all that, we can go through the example here.
02:35
So a, give an example of a sequence which verkonjas but does not converge.
02:41
I'm going to set xn to be equal to sine of n.
02:48
I'm going to note that by having a value, if epsilon equals 1, we have for all values of n, xn minus 0 in particular is always going to be less than or equal to 1.
03:10
Because sine of n, for real n indeed, will always be in the integral 0 to 1.
03:20
That means that for all natural n, xn is going to be within 1 of 0, which means that it verkonjas to 0.
03:27
Check.
03:28
But it's plain to see that it doesn't converge.
03:31
I'm not going to provide a particular argument for that, but sine of n does not converge to anything as a sequence over natural n.
03:45
Now b, i'd like to show that if the limit, or perhaps the millet, of a sequence exists, then it is not unique.
03:54
In fact, if it exists, then it's not unique.
03:57
Well, let's see here.
04:00
Let's suppose that xn is a sequence that verkonjas to x...