Thread: Proof: Limits of Sequences

Supply a proof for the following theorem: Suppose that f is continuous at a and that x(subscript)n tends to a as n goes to infinity. Then there is an integer N such that f(x(subscript)n) is defined for all integers n > N; furthermore, f(x(subscript)n) tends to f(a) as n goes to infinity.

Should I be typing these in LaTeX and posting them here somehow? I hope it is obvious that typing this out with regular script that x(subscript)n represents the sequence x-sub-n. This is my first post so I hope to clean up my future posts before submission to aviod confusion or ambiguity.

[tex]\lim_{n \to \infty}(x_n)\to a[/tex] gives

Originally Posted by paulread

Supply a proof for the following theorem: Suppose that f is continuous at a and that x(subscript)n tends to a as n goes to infinity. Then there is an integer N such that f(x(subscript)n) is defined for all integers n > N; furthermore, f(x(subscript)n) tends to f(a) as n goes to infinity.

Should I be typing these in LaTeX and posting them here somehow? I hope it is obvious that typing this out with regular script that x(subscript)n represents the sequence x-sub-n. This is my first post so I hope to clean up my future posts before submission to aviod confusion or ambiguity.

Let be any neighborhood of , by 's continuity we know that is a neighborhood of and so by assumption all but finitely many of and so all but finitely many of are in .

The conclusion follows.

P.S. You should probably specify what you kind of space we are in here. The above works for any topological space with a convergent sequence. Some weird things happen though. For example, any sequence in an indiscrete space converges to every point.

By the way, some textbooks define "f is continuous at x=a" to mean that for every sequence that converges to a.