Let $\{x_n\}$ be a sequence in a Hilbert space $H$ and $a$ is an element of $H$, Given that $$\underset{n \to \infty}{\lim {\sup}}\ \|x_n -a\| \leq \|x^*-a\| $$ where $x^*$ is some fixed element of $H$. Then how to prove that $\{x_n\}$ is a bounded sequence.
I started with $$\|x_n\| \leq \|x_n-a\|+ \|a\|$$Then unable to solve further. How to make use of given inequality. Can i write $$\|x_n\| \leq \underset{n \to \infty}{\lim {\sup}}\ \|x_n -a\|+ \|a\|$$ Please explain this.