There are the definitions which we need for the proof of the theorem :
There is the theorem: If ${f_n}$ is a Cauchy sequence in $\mathscr L^2(\mu)$ , then there exists a function $f$ $\in$ $\mathscr L^2(\mu)$ such that ${f_n}$ converges to $f$ in $\mathscr L^2(\mu)$ .
This says, in other words, that $\mathscr L^2(\mu)$ is a complete metric space.
There is the proof:
since ${f_n}$ is a Cauchy sequence, we can find a sequence ${n_k}$, $k$ $=$ $1,2,3,...,$ such that
$||f_{nk}-f_{n{k+1}}||$ $\lt$ $\frac {1}{2^k}$. ($k$ $=$ $1,2,3,...,$).
Choose a function $g$ $\in$ $\mathscr L^2(\mu)$ . By the Schwarz inequality,
$\int_{X} |g(f_{nk}-f_{n{k+1}})| d\mu $ $\leq$ $\frac {||g||}{2^k}$ .
Hence
$\sum_{k=1}^\infty $ $\int_{X} |g(f_{nk}-f_{n{k+1}})| d\mu $ $\leq$ $||g||$.
I couldn't understand How do we get the first inequality by the Schwarz inequality, and thereby, I also couldn't understand how do we get the last inequality ( $\sum_{k=1}^\infty $ $\int_{X} |g(f_{nk}-f_{n{k+1}})| d\mu $ $\leq$ $||g||$.)
Any help would be appreciated.