Existence of convergent sequence implies convergence of function as $t\to\infty$

Question

Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in C([0,\infty);H_{0}^{1}(\Omega))$. Assume there exists a sequence $\{t_{n}\}_{n\in\mathbb{N}}$ such that $t_{n}\to\infty$ and $||u(t_{n})||_{H_{0}^{1}(\Omega)}\to 0$ as $n\to\infty$. Is it possible to show that $||u(t)||_{H_{0}^{1}(\Omega)}\to 0$ as $t\to\infty$? I am not sure how to show this if this is possible. If this is not possible, then is there any counter example? Moreover, can I still obtain similar result if I replace $H_{0}^{1}(\Omega)$ with $X$ a Banach space?

Any hint is much appreciated! Thank you!

Answer 1

Let $u(t)=\sin(t\pi)f$ where $f$ is a fixed non-zero element of $H_0^{1}(\Omega)$. Take $t_n=n$.