Understanding f(y) in a Nonempty Set X of $\mathbb{R}^n$ - Bob19

[Bob19]

Hi

i have this following assignment in Analysis

Given $$X \subseteq \mathbb{R}^n$$ which is a nonempty subset of $$\mathbb{R}^n$$

The set $$\{ \| | x -y \| | \ | x \in X \}$$ has an infimum such that

$$f(y) = \{ \| | x -y \| | \ | x \in X \}$$

where $$f: \mathbb{R}^n \rightarrow \mathbb{R}^n$$

I need a hint on howto show that if $$y \in X$$ then f(y) = 0 ??

Regards,

Bob19

 

[benorin]

Fix x in X. What is the shortest distance between x and y if y is allowed to be in X (note that x is in X)?

 

[matt grime]

very similar to mathboy20s post in this subforum