Convergence of a sequence in $L^2$

Question

Let $\{a_k\}_{k\ge 0}$ a bounded sequence in $L^2(\Omega)$ and $a \in L^2(\Omega)$. I can prove the following inequality:

$\left\lVert a_k - a \right\rVert_{L^2(\Omega)}^2 \le \left\lVert a_{k-1} - a \right\rVert_{L^2(\Omega)}^2 - \left\lVert a_{k} - a_{k-1} \right\rVert_{L^2(\Omega)}^2$,

Then we have:

$\left\lVert a_k - a \right\rVert_{L^2(\Omega)}^2 \le \left\lVert a_{k-1} - a \right\rVert_{L^2(\Omega)}^2$ and $\sum_{k=1}^\infty \left\lVert a_{k} - a_{k-1} \right\rVert_{L^2(\Omega)}^2 \le \left\lVert a_{0} - a \right\rVert_{L^2(\Omega)}^2$.

Using these inequalities, is there a way to prove that :

$\lim_{k\rightarrow \infty} \left\lVert a_{k} - a \right\rVert_{L^2(\Omega)} = 0$ ?

Answer 1

Easy counter-example: Let $a_1=a_2=\cdots$ but $a \neq a_1$. Then the inequality holds but the conclusion fails.