I'm trying to understand a paragraph from the article Christ, Kiselev: Maximal functions associated to filtrations. After the proof of Theorem 1.1., the authors deduce a theorem of Menshov as a corollary:

Let $(X, \mu)$ be a measure space and let $(\phi_n)_n$ be an orthonormal sequence in $L^2(X)$. Let $1 \le p < 2$. Then for every sequence $(c_n)_n \in \ell^p$ the series $$\sum_{n=1}^\infty c_n\phi_n$$ converges a.e. in $X$.

Obviously the series $\sum_{n=1}^\infty c_n\phi_n$ converges in $L^2(X)$ due to Parseval since in particular we have $(c_n)_n \in \ell^2$.

Theorem 1.1. from the article gives that the (sublinear) map $$\ell^p \to L^2(X), \quad c = (c_n)_n \mapsto \sup_{N\in\Bbb{N}} \left|\sum_{n=1}^N c_n\phi_n\right|$$ is well-defined and bounded in the sense that there exists a constant $C>0$ such that $$ \left\|\sup_{N\in\Bbb{N}} \left|\sum_{n=1}^N c_n\phi_n\right|\right\|_{L^2(X)} \le C\|(c_n)_n\|_p, \quad \text{ for all }(c_n)_n \in \ell^p.$$

The authors now conclude that the statement of the theorem clearly follows from this fact and from the fact that $\sum_{n=1}^\infty c_n\phi_n$ obviously converges a.e. when $(c_n)_n$ is finitely-supported.

I'm not sure how it follows. I noticed that for $M \ge N$ the Cauchy sums are also bounded by the supremum by reverse triangle inequality: $$\left|\sum_{n=N+1}^M c_n\phi_n\right| \le 2\sup_{K\in\Bbb{N}} \left|\sum_{n=1}^K c_n\phi_n\right|.$$ So this is bounded a.e., however we would need that it goes to $0$ as $M,N\to\infty$. Am I missing something easy?