# Proving The Continuous Theorem for Sequences

by Lucretius
Tags: continuous, proving, sequences, theorem
 P: 159 1. The problem statement, all variables and given/known data 4.8 Show the following continuous theorem for sequences: if $$a_n \rightarrow L$$ and f is a real valued function continuous at L, then $$bn = f(a_n) \rightarrow f(L)$$. 2. Relevant equations No real relevant equations here. Just good old proof I'm thinking. 3. The attempt at a solution Well, I stared at this for an hour today. I was able to complete the rest of the assignment but this one has me stumped. I realize that $$\displaystyle\lim_{n\rightarrow\infty}a_n=L$$ and that for a real-valued function to be continuous at L that $$\displaystyle\lim_{x\rightarrow x_0}f(x)=f(x_0)=L.$$ I don't know what to do from here though. How do I get f(L) from f(x0)=L, and then get f(a_n) from just plain old a_n. This thing makes intuitive sense to me; it's blatantly obvious it's right - proving it has ... well.. proven to be really hard!
 Sci Advisor HW Helper P: 2,537 Proving The Continuous Theorem for Sequences It looks like you might be getting confused by using two different $L$'s. Typically, the $a_n$ and $b_n$ will not converge to the same value. Really, this is straightforward stuff... given some $\epsilon$ greater than $0$ can you show that there is some $N$ so that $n>N \Rightarrow |f(a_n)-f(L)| < \epsilon$?
 Sci Advisor HW Helper P: 2,537 It's a little abstract. Do you know if there's a $\delta$ so that: $$|a_n-L|<\delta \Rightarrow |f(a_n)-f(L)| < \epsilon$$