Uniformly continuity of a real valued function $f$ 
Let $f:\mathbb{R}\to \mathbb{R}$ be a function given by $f(x)=\sum_{k=1}^\infty \frac{g(x-k)}{2^k}$ 
  where $g:\mathbb{R}\to \mathbb{R}$ is a uniformly continuous function such that the series converges for each $x$ belongs to $\mathbb{R}$.
  Then show that $f$ is uniformly continuous.

How I show this. Please help me to solve this.

If $g$ is uniformly continuous in $\mathbb{R}$ then given $\epsilon>0$ there exist $\delta>0$ such that if $|x-y|<\delta$ then $|g(x)-g(y)|<\epsilon$.
Hence, if $|x-y|<\delta$, then $|(x-k)-(y-k)|=|x-y|<\delta$ and
$$|f(x)-f(y)|\leq \sum_{k=1}^\infty \frac{|g(x-k)-g(y-k)|}{2^k}\leq 
\sum_{k=1}^\infty \frac{\epsilon}{2^k}=\epsilon,$$
that is $f$ is uniformly continuous in $\mathbb{R}$.
