Let $T>0$ and $$f_n : [0,T] \rightarrow \mathbb{R} $$ be a sequence of continuous functions. Furthermore, let $$ \sum_{k = 0}^{\infty} \sup_{t \in [0,T]} \lVert f_{k+1}(t) - f_{k}(t) \rVert < \infty. $$
Does this already imply, that there exists a continuous function $$f : [0,T] \rightarrow \mathbb{R} $$ such that $(f_n)$ converges uniformly to $f$ on $[0,T]$?
Yes. For $n <m$ we have $|f_n(t)-f_m(t)| \leq |f_n(t)-f_{n+1}(t)|+|f_{n+1}(t)-f_{n+2}(t)|+\cdots+|f_{m-1}(t)-f_m(t)|$. Can you verify from this that $\{f_n\}$ is uniformly Cauchy. This would then show that $f_n$ converges uniformly to a function $f$ which is necessarily continuous.