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I'm really confused with working with sequences of functions and their sup.

Let A be a non-empty subset of $\mathbb{R}$. For all $n \in \mathbb{N}$ and let $f_n : A \to\mathbb{R}$ be a continuous function.

Suppose that for all $x \in A$ the sequence $f_1(x), f_2(x)...$ is bounded.

Define the function $g : A \to\mathbb{R}$ by: $g(x)=\sup\{f_1(x), f_2(x)...\}$

Then if we have some $a \in A$ and $\epsilon\gt0$, how would one go about proving that there is a $\delta\gt0$ such that, for all $x\in A$ with $|x-a|\lt\delta$ one has that $g(a)-g(x)\lt \epsilon$ ?

Additionally, how can we show whether the function $g$ is continuous?

I'm trying to do a standard epsilon-delta proof for the first part, but can't seem to introduce an $|x-a|$ term into the working.

drhab
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Analysis is fun
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1 Answers1

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Hint: $$g(x) - g(a) \leqslant \sup g(x) - \inf g(a).$$ Use the fact: $\sup A-\inf B=\sup(A-B).$

jinx
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  • So, $g(x)-g(a) \leq sup g(x) - inf g(a) = sup(g(x-a)) \lt sup(g(\delta))$? – Analysis is fun Oct 03 '17 at 15:16
  • no, its $\sup(g(x)-g(a))$ and now use the continuity of $f_{n}$ – jinx Oct 03 '17 at 15:18
  • This hint is not helpful, in my opinion. To Analysis is fun: it might be wise to check out this https://math.stackexchange.com/questions/630339/pedagogy-how-to-cure-students-of-the-law-of-universal-linearity question, and then reevaluate the manipulations in your comment. – gj255 Oct 03 '17 at 15:24