Is every normed space $X=\cup_{n=1}^{\infty} B(0, n)$? Why?

Question

Is every normed space $X=\cup_{n=1}^{\infty} B(0, n)$? Why?

How can one prove that the union of such balls encompasses any kind of normed space?

Answer 1

If $x\in X$, take $n\in\mathbb N$ such that $n>\|x\|$. Then $x\in B(0,n)$.