Let be $X$ a closed and bounded set of $\mathbb{R}^n$: how to prove that there exsist $a,b\in\mathbb{R}$ such that $X\subseteq[a,b]^n$?
As reference I said that I know that if $X$ is bounded then there exist some $x_0\in\mathbb{R}^n$ and some $r>0$ such that $X\subseteq B(x_0,r)$.
Could someone help me, please?
Let's start with there existing $x_0$ and $r>0$ such that $X\subseteq B(x_0,r)$. Let $(y_1,\ldots,y_n)$ be the coordinates of $x_0$. Then $$B(x_0,r)\subseteq [y_1-r,y_1+r]\times [y_2-r,y_2+r]\times\cdots \times [y_n-r,y_n+r]$$ This is because any point $z$ with $\|z-x_0\|<r$, meaning $$\sum_{i=1}^n(z_i-y_i)^2<r^2$$ must have that $(z_i-y_i)^2<r^2$ for all such $i$ since these quantities are all nonnegative. Thus $z_i\in [y_i-r,y_i+r]$.
In order to get this to be in $[a,b]^n$ for some $a,b$, let $a$ be the minimum of $y_i-r$ over all $i$ and let $b$ be the maximum of $y_i+r$ over all $i$.
