$(C(A,\mathbb{R}),\|\cdot\|_\infty)$ is a Banach Space when $A$ is Compact

Question

For a normed vector space $(V,\|\cdot\|_{V})$, let $A\subseteq V$ be compact, and let $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$ be the space of continuous functions from $A$ to $\mathbb{R}$ with respect to the sup-norm. How can I use specifically the Weierstrass M-Test, and facts about absolutely convergent series in normed vector spaces to show that $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$ is a Banach Space?
Can I assume that I have an arbitrary absolutely convergent series $\Bigl(\sum_{k=0}^n f_{k}\Bigr)_{n\in\mathbb{N}}$ in $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$, and then just let $M_n=\|f_n\|_{\infty}$ and then apply the M-Test to show that the sequence converges?