Justifying Set Boundedness of $S_{||\cdot||_2}$ in $\mathbb{R}^n

[evinda]

Hello! (Wave)We have that $S_{||\cdot||_2}:= \{ x \in \mathbb{R}^n: ||x||_2=1\}$.

How can we justify that the above set is bounded?

Do we just say that if $x \in S_{||\cdot||_2}$ then $||x||_2=1 \leq 1$ and so the set is bounded. How could we justify it more formally?

 

[Euge]

Hi evinda,

To show formally that this set is bounded, you need to prove that there is a positive number $K$ such that for all $x,y\in S$, $\|x-y\|_2\le K$. Using the triangle inequality you'll find that $K=2$ is a suitable upper bound.