If I know that $\sum_M^\infty v(x)$ converges towards $f$, then $$\left| \left( f + \sum_{1}^{M-1} v(x) \right) - \sum_{1}^N v(x)\, \right| = \left|\;f - \sum_{M}^N v(x)\,\right| < \epsilon$$ for large enough $N$, and so our general series converges towards $f + $ the terms from 1 to $M-1$.
What is wrong with this proof? It is not mentioned in the book, and they instead opt for a more complicated strategy, leading me to think something is fishy about above.