A question about Riesz - Fischer theorem's proof

Question

In Riesz-Fischer theorem's proof, when we put $$ g_k =|f_{n_1}|+|f_{n_2}-f_{n_1}|+ \cdots + |f_{n_k}-f_{n_{k-1}}| $$ it is easy to get (by Minkowski's inequality) $$ \left \| g_k \right \|_p \leq \left\|f_{n_1}\right\|_p+\left\|f_{n_2}-f_{n_1}\right\|_p+ \cdots + \left\|f_{n_k}-f_{n_{k-1}}\right\|_p. $$

Can you show me how to formally pass to the limit in order to obtain

$$ \left \| g \right \|_p \leq \sum_{k=1}^\infty \left \| f_{n_k}-f_{n_{k-1}} \right \|_p $$

? Thanks

Answer 1

Since for all $k$ we have

$$\lVert g_k\rVert_p \leqslant \sum_{m=1}^\infty \lVert f_{n_m} - f_{n_{m-1}}\rVert_p$$

(with $f_{n_0} = 0$), and $\lvert g_k\rvert^p \uparrow \lvert g\rvert^p$ monotonically, the monotone convergence theorem tells us

$$\lVert g\rVert_p \leqslant \sum_{m=1}^\infty \lVert f_{n_m} - f_{n_{m-1}}\rVert_p.$$