The problem is a sort of Cauchy-Schwarz inequality:
Let $(x_n)_{n\in\mathbb{N}}$ be a sequence in a Hilbert Space $H$, and $(c_n)_{n\in\mathbb{N}}\in l^2(\mathbb{N}).$ Let also $F$ be a finite subset of $\mathbb{N}.$
We know that $\exists C>0$ such that for every $x\in H$ we have the property $\sum_n |(x_n,x)|^2\le C\|x\|^2.$
Prove that $$\big\|\sum_{n\in F}c_nx_n\big\|^2\leq \sup_{y\in H, \|y\|=1} \big(\sum _{n\in F}c_n^2\big)\big(\sum_{n\in F}|(y,x_n)|^2\big),$$ where $(\cdot,\cdot)$ denotes the scalar product.
This is quite similar to this question, but actually seems to be its inverse.
The right term reminds a lot of an operator norm but I couldn't define any function that could be useful for the problem. Also, splitting the left term into scalar products leads to some matrix-connected calculation that didn't seem to be useful.
Hint: for any $y\in H$, consider $$ \Big(y,\sum_{n\in F}c_n x_n\Big)=\sum_{n\in F}c_n(y,x_n). $$ Apply the C-S to the right hand side and take the supremum over $y$ in the unit ball.