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

Show that if f: S -> R^{n}is uniformly continuous and S is bounded, then f(S) is bounded.

2. Relevant equations

Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e

bounded: a set S in R^{n}is bounded if it is contained in some ball about the origin. That is, there is a constant C s.t. |x|<C for every x in S.

3. The attempt at a solution

I understand the idea of the proof pretty well but I cannot write the correct mathematical interpretation of it down.

Basically, S is bounded, so it can be divided into segments (TA called them partitions which is confusing since S is in R^{n}not R).

Each segment can be made smaller than d. Then, by uniform continuity we know that f(segment) is smaller than e (bounded) so we can draw a ball around it.

Since there is a finite number of segments, there are a finite number of balls f(segment). Hence, we can draw a bigger ball around all of them, and thus f(S) is bounded.

How do I put this into math symbols?

Thanks for your help =)

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# Homework Help: Uniform continuity, bounded subsets

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