Let $(X,\rVert\cdot\lVert)$ is a Banach space and $\{x_n\}\subseteq X$ a sequence. If I prove that $\{x_n\}$ is bounded that is $$\sup_n\rVert x_n\lVert<\infty\quad\forall n\in\mathbb{N},$$Can I conclude that the sequence $\{\lVert x_n\rVert\}_n$ is also bounded?
Thanks!
Yes and if you want a justification here it is. $\{ \| x_n \| \}_n$ is a sequence of real numbers, if it's easier to think of it: $\{ y_n \}_n$ where $y_n=\| x_n \|$. So for it to be bounded, need $$ \sup_n |y_n| < \infty $$ which is true since $\| x_n \| \geq 0$ hence the absolute value of $\| x_n \|$ is equal to $\| x_n \|$, hence $$ \sup_n |y_n| = \sup_n \| x_n \| < \infty $$