Let $(V_1, \lVert \cdot \rVert_{V_1})$ be a normed vector space and $(V_1, \lVert \cdot \rVert_{V_2})$ be a complete normed vector space. Let $(C_b(V_1, V_2), \lVert\cdot\rVert)$ be the vector space of all continuous bounded functions $f:V_1\to V_2$. Prove this version of the Weierstrass M-Test:
If $\{f_k\}$ is a sequence in $C_b(V_1, V_2)$ where $\lVert\ f_k \rVert_\infty\le M_k$ for all $k$, and $\left\{ \sum\limits_{k=1}^n M_k \right\}_{n=1}^\infty$ converges to $\sum\limits_{k=1}^\infty M_k$ in $\mathbb{R}$, then $\left\{ \sum\limits_{k=1}^n f_k \right\}_{n=1}^\infty$ converges to $\sum\limits_{k=1}^\infty f_k$ uniformly.
But isn't it always the case that, even if $\sum\limits_{k=1}^\infty M_k$ is divergent, $\left\{ \sum\limits_{k=1}^n M_k \right\}_{n=1}^\infty$ will converge to $\sum\limits_{k=1}^\infty M_k$? I seem to misunderstand what exactly this statement means. Or maybe this statement is lacking some details?
I will, nevertheless, try to prove this (making my own interpretation of the statement above).
I'm not sure how to prove this statement. Would appreciate some basic outline.