I have posted the following question recently
Why do uniformly continuous functions form a Banach space (with the sup norm)?
I know now that space of uniformly continuous bounded functions with the sup norm is a Banach space (since it's complete).
But how can I prove that the limit of a Cauchy sequence in this space is indeed uniformly continuous.
The pointwise limit has to be the limit of a Cauchy sequence of uniformly continuous bounded functions. However I am not able to the fact that it is uniformly continuous. Please help me with this, (I am desperate).
The definition of uniform continuity is:
for every $\epsilon$ there exists a $\delta>0$ such that $|g(x)-g(y)|<\epsilon$ for all $x,y\in A$ s.t. $d(x,y)<\delta$.