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

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The set $S_{||\cdot||_2} = \{ x \in \mathbb{R}^n: ||x||_2=1\}$ is bounded because all elements have a fixed norm of 1. To justify this formally, one must demonstrate that there exists a positive constant $K$ such that for any two points $x, y \in S_{||\cdot||_2}$, the distance $\|x-y\|_2$ is less than or equal to $K$. By applying the triangle inequality, it can be shown that $K=2$ serves as an appropriate upper bound for the distances between points in the set. Thus, the boundedness of $S_{||\cdot||_2}$ is established.
evinda
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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?
 
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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.
 
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