(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

4.8 Show the following continuous theorem for sequences: if [tex]a_n \rightarrow L[/tex] and f is a real valued function continuous at L, then [tex]bn = f(a_n) \rightarrow f(L)[/tex].

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 [tex]\displaystyle\lim_{n\rightarrow\infty}a_n=L[/tex] and that for a real-valued function to be continuous at L that [tex]\displaystyle\lim_{x\rightarrow x_0}f(x)=f(x_0)=L.[/tex] 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!

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# Proving The Continuous Theorem for Sequences

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