00:01
Hello, so here to determine the convergence of the sequence, so we have x of n to be equal to 1 over n, in the finite complement topology on r.
00:11
So the finite complement topology on r, we have the open sets are those whose complements are finite, which implies that a set is closed.
00:21
So a set is closed if and only if, if and only if, iff means if and only if it is finite.
00:35
So, a sequence then xn converges to a limit l in a topological space.
00:41
If for every open set mu, let's say containing l, there exists a natural number n, such that for all little n greater than or equal to that big number n, we have that x sub n is an element or is in the open set mu.
00:58
Into u.
00:59
So we examine the behavior of x sub n equals 1 over n as n approaches infinity.
01:05
What we have here for our case of l equals 0, we can consider any set u containing 0 in the finite complement topology.
01:13
And the complement of u being finite means that eventually x sub n equals 1 over n will lie within u for sufficiently large n, because only a finite number of values u's of 1 over n can fall outside of u...