Strong convergence on the unit sphere of $l_2$ Let $(p_n)$ be a strictly  increasing sequence of natural numbers and $(\epsilon_n)$ a positive  sequence decreasing to $0$. Suppose $x_n$ is a sequence in $S(l_2)$ (the unit sphere of $l_2$) with the following property: For any $j\in\mathbb{N}$, and for any $1\leq k\leq j$
 $$
\sum_{i>p_k}|(x_j,e_i)|^2<\epsilon_k
$$
Can I find a subsequence of $(x_n)$ that converges (strongly) to some $x\in S(l_2)$? 
 A: Yes. After passing to a subsequence, assume $x_k \rightharpoonup y$ weakly as $k \to \infty$. Note this implies $\langle x_k, e_j \rangle \to \langle y, e_j \rangle$ as $k \to \infty$ for all $j \in \mathbb{N}$. The property given ensures some form of uniform decay at $\infty$, so we will have strong convergence using a standard argument.
Fix $\varepsilon > 0$. Let $K$ be large enough that $k \geq K$ implies $\varepsilon_k < \varepsilon$ and $\sum_{j=p_K+1}^\infty \lvert \langle y, e_j \rangle \rvert^2 < \varepsilon$. Let $N \geq K$ be so large that $\lvert{\langle x_k, e_j \rangle - \langle y, e_j \rangle}\rvert^2 < \varepsilon/p_K$ for $1 \leq j \leq p_K$ and $k \geq N$. We have
$$ \begin{aligned}
\sum_{j=1}^\infty \lvert{\langle x_k, e_j \rangle - \langle y, e_j \rangle}\rvert ^2 &\leq \sum_{j=1}^{p_K} \lvert{\langle x_k, e_j \rangle - \langle y, e_j \rangle}\rvert ^2 + \sum_{j=p_K + 1}^\infty \lvert{\langle x_k, e_j \rangle}\rvert ^2 + \sum_{j=p_K + 1}^\infty \lvert{\langle y, e_j \rangle}\rvert ^2 \\ &\leq \varepsilon + \varepsilon_K + \varepsilon \leq 3 \varepsilon
\end{aligned} $$
for all $k \geq N \geq K$.
