Power series convergence in boundary problem

Say I have a power series $\sum_{k=0}^\infty a_k x^k$ which converge uniformly on $\left[0, 1\right)$ . Now I need to prove that series $\sum_{k=0}^\infty a_k$ are convergent.

My idea is to use equivalent Cauchy form $\forall \epsilon\ \exists N \text{ such that }\sup_\left[0, 1\right) |S_n(x)-S_m(x)|<\epsilon\quad \forall m,n\ge N\ $ where $S_n = \sum_{k=0}^n a_k x^k$. Because of continuity of $P(x) = |S_n(x)-S_m(x)|$ we see that $\sup_\left[0, 1\right) P(x) = \sup_\left[0, 1\right] P(x)$ and this way it can be proved that $\sum_{k=0}^\infty a_k$ convergent as Cauchy sequence.