Problem: If a series $\sum_{k=0}^\infty g_k$ of functions satisfies the Cauchy criterion uniformly on a set $S$, then the series converges uniformly on $S$.
If the series satisfies the Cauchy criterion uniformly on $S$, then
$$\forall \epsilon>0, \ \exists N : n \geq m \geq N \Rightarrow \bigg|\sum_{k=m}^n g_k(x) \bigg| < \epsilon , \forall x \in S$$
Therefore, we can define
$$f_l(x) = \sum_{i=0}^l g_i(x)$$
Which transforms the criterion into
$$\bigg| \sum_{k=0}^ng_k(x) - \sum_{k=0}^m g_k(x) \bigg| = |f_n(x) - f_m(x)| < \epsilon$$
Therefore, the definition of uniformly Cauchy is satisfied.
Is this correct? Is the converse true?
If you write $f_l(x) = \sum_{i=0}^l g_i(x)$ instead of $f_l(x) = \sum_{i=0}^l f_i(x)$, then everything is correct.
And yes, the converse is also true. Try a proof !