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Let $$f(x) = \sum_{n = 1}^\infty \frac{g(x-n)}{2^n}$$ where $g$ is a uniformly continuous function such that the series converges for each $x$.

We need to show that $f:\mathbb{R}\rightarrow\mathbb{R}$ is uniformly continuous. Can any one just tell me in which way I should try? From the definition I guess it will be bit clumsy.

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What is the function $g$? – Mhenni Benghorbal Jan 4 '13 at 6:42
@MhenniBenghorbal $g$ is uniformly continuous, in my original post it was there. – Un Chien Andalou Jan 4 '13 at 6:46
@Kuttus: Sorry about my mis-edit. I fixed it. – mixedmath Jan 4 '13 at 6:48
@mixedmath :) .. – Un Chien Andalou Jan 4 '13 at 6:48
up vote 2 down vote accepted


You can proceed using only the definition of uniform continuity. In addition, I might want to add that $\sum_{n = 1}^\infty \frac{A}{2^n} = A$.

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