If $(V,\| \cdot\|)$ is a real normed space. How can I proof that if $V$ is not the trivial space then it is not bounded.

Question

I was studying normed spaces and I got into trouble with these two questions that I can't answer: Let $(V, \| \ \|)$ be a real normed space and I want to know if the next statement is correct and how can I prove it: If $V$ is not the trivial space then $V$ is not bounded. And according to that how can I explain that if in a non-trivial normed space we defined a distance $d$ in such that the space results bounded then this distance $d$ can not be induced by a norm. i.e it is impossible there exists a norm $\| \ \|$ in the space such that $d(x,y)=\|y-x\|$

Answer 1

If $x\in V\setminus\{0\}$, then $\lim_{n\to\infty}\lVert nx\rVert=\lim_{n\to\infty}n\lVert x\rVert=\infty$.