Problem:
Let $X\subset \mathbb{R}^{n}$ be a compact set. Prove that the set $Y=\left \{ y\in \mathbb{R}^{n}: \left | x-y \right |=2000 : x\in X \right \}$ is compact.
First, I don't understand how the absolute value of the difference between x and y: $\left | x-y \right |$ be a number in $\mathbb{R}$, akthough $x$ and $y$ are from $\mathbb{R}^{n}$.
Second, there many definitions for compact set(closed and bounded, every sequence in the set has a convergent subsequence in the set, every open cover has a finite subcover). I am trying to use the first definition, any help please?