Let $H$ be a Hilbert space, $\left(a_n\right)$ is a bounded sequence in $H.$ Then there exists a subsequence $\left(a_{n_k}\right)$ of $\left(a_n\right)$ which converges weakly in $H.$ On the other hand, $\left\Vert a_{n_k}\right\Vert$ is bounded in $\mathbb{R},$ thereby it follows from the Bolzano-Weierstrass theorem that there exists a subsequence $\left(a_{n_{k_m}}\right)$ of $\left(a_{n_k}\right)$ such that $\left\Vert a_{n_{k_m}}\right\Vert$ is convergent in $\mathbb{R}.$ Besides, $\left(a_{n_{k_m}}\right)$ is also a weakly convergent sequence. Therefore, $\left(a_{n_{k_m}}\right)$ is a convergent sequence in $H.$
I can't find any mistake in this proof. Please show me a mistake? Thank you!