# Showing $\sum_{n = 1}^\infty \frac{g(x-n)}{2^n}$ is uniformly continuous

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. –  El Angel Exterminador Jan 4 '13 at 6:46
@Kuttus: Sorry about my mis-edit. I fixed it. –  mixedmath Jan 4 '13 at 6:48
@mixedmath :) .. –  El Angel Exterminador Jan 4 '13 at 6:48

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