- #1
julypraise
- 110
- 0
Homework Statement
Could you please check the following calculation is right?
Let X be a metric space, and A its nonempty subset.
Define [itex]\inf_{a \in A} d(x,a) = d(x,A) [/itex] for any x in X
We have the following facts (don't have to check this)
If [itex]a[/itex] is in the closure of [itex]A[/itex] then d(a,A)=0.
So my calculuation is as follows. Let x be in X and a_0 be in the closure of A
Then
[itex] d(x,a_{0}) \leq \inf_{a\in A} \{ d(x,a) + d(a,a_{0}) \} = \inf_{a\in A}d(x,a) + \inf_{a\in A}d(a,a_{0}) = d(x,A)[/itex]
Homework Equations
The Attempt at a Solution
The calculation looks somehow right (probably because of my bad). I have to show the existence of such a_0. In R^2 dimension, I can draw this and get it, but such a point a_0 is not arbitrary in that case whereas the a_0 in my calculation is arbitrary and works fine which is very stupid...
Probably the equality made in the mid is not right... But if it is not right, how can I show the existence of such a_0?