# Thread: Proof: Limits of Sequences

1. ## 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.

2. $$\lim_{n \to \infty}(x_n)\to a$$ gives $\lim_{n \to \infty}(x_n)\to a$

Let $N$ be any neighborhood of $f(x)$, by $f$'s continuity we know that $f^{-1}(N)$ is a neighborhood of $x$ and so by assumption all but finitely many of $\left\{x_n\right\}_{n\in\mathbb{N}}$ and so all but finitely many of $f(x_n)$ are in $ff^{-1}(N)\subseteq N$.
4. By the way, some textbooks define "f is continuous at x=a" to mean that $\lim_{n\to\infty} f(x_n}= f(a)$ for every sequence $\{x_n\}$ that converges to a.