# 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 $\displaystyle \lim_{n \to \infty}(x_n)\to a$

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