How to prove that for a compact set to be measure zero, we need $\forall \epsilon > 0$, there exists a finite covering satisfying the necessary condn

Question

In the book of Mathematical Analysis by Zorich, it is given that

A compact set $K \subseteq \mathbb{R}^n$ is a set of measure zero iff $\forall \epsilon > 0$, there exists a finite covering of $K$ s.t the sum of the measures of the intervals of this covering is less than $\epsilon$.

I can prove the $\Rightarrow$ part, but I'm having trouble show the reverse, I mean how does being able to find a finite covering for $\epsilon$ implies that for any open cover of $K$, we can find a finite covering ?

Answer 1

It seems you are trying to prove compactness. The thing is: you do not have to!

View the theorem like this

$K$ is compact $\;\Rightarrow\;$ ($K$ is of measure zero $\;\Leftrightarrow\;$ $\exists$ finite cover with summed measure $\le\epsilon$)

So compactness is a common assumption. You only have to prove $\Leftarrow$ and $\Rightarrow$ inside the paranthesis. No arrow points to compactness.