# Uniform Cauchyness

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 $$|f_n|$$.

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.