Uniform Cauchyness

Question

Assume the sum of $|f_n|$ is uniformly cauchy. Does this imply that the sum of $(f_n)$ is uniformly cauchy?

My reasoning is yes, since every $f_n$ is bounded by $|f_n|$.

Answer 1

Yes since $$ \lVert \sum_{n=k+1}^m f_n \rVert\leq \sum_{n=k+1}^m \lVert f_n \rVert=\sum_{n=k+1}^m \lVert |f_n| \rVert\to 0 $$ as $m\to \infty$ and $k\to \infty$ where $\lVert\cdot \rVert$ is the uniform norm.