Is there anything wrong with the following:
If $\{f_n\}$, $\{g_n\}$ are two sequences of functions in a Hilbert space $H$, then
$\begin{align*} \sqrt{\sum_n |\langle f,f_n \rangle|^2} - \sqrt{\sum_n |\langle f,f_n - g_n \rangle|^2}&\leq \sqrt{\sum_n |\langle f,g_n \rangle|^2}\\ &\leq \sqrt{\sum_n |\langle f,f_n \rangle|^2} + \sqrt{\sum_n |\langle f,f_n - g_n \rangle|^2} \end{align*}$
for all $f\in H$. Thanks!