- #1

- 165

- 1

**1. Let M be a compact metric space. If r>0, show that there is a finite set S in M such that every point of M is within r units of some point in S.**

**2. Relevant theorems & Definitions:**

-Every compact set is closed and bounded.

-A subset S of a metric space M is sequentially compact if every sequence in S has a subsequence that converges to a limit in S.

-If every open covering of S reduces to a finite subcovering then we say that S is covering compact.

-Every compact set is closed and bounded.

-A subset S of a metric space M is sequentially compact if every sequence in S has a subsequence that converges to a limit in S.

-If every open covering of S reduces to a finite subcovering then we say that S is covering compact.

**3. Attempt at the solution**

To begin with I don't completely understand coverings. So I am unsure whether I should use the sequential definition or the covering definition. Could someone please explain coverings?

I also wondered if S is a subset of a compact metric space, does it follow that S is compact?

INFORMAL PROOF: Since M is compact, it follows that every sequence in M has a convergent subsequence in M. Need to show that for any point in M is within some distance from a point in S. Let [tex]m_{k_{j}}[/tex] be such a subsequence. So for [tex]\epsilon>0[/tex] there exists N such that [tex]m_{k_{j}}[/tex] is within [tex]\epsilon[/tex] units of some point, call it m.

Not sure where to go from here.