How can we show that any finite cover of A also covers the interval (0,1)?

[lmedin02]

Homework Statement

Let A be the set of all rational numbers between 0 and 1. Show that for any "finite" collection of intervals $$I_n$$ that cover A the following inequality holds: $$\sum I_n \geq 1$$.

Homework Equations

We are using the definition of the outer measure here. Where the outer measure of A is define as the infimum of $$\sum I_n$$ where the infimum is taken over all possible open intervals that cover A.

The Attempt at a Solution

I know that the outer measure of A is 0 because A is a countable set. If I consider finite covers of A, then the sum of their lengths obviously add up to 1 or greater. But I still have no sense of direction on where to continue with this problem.

 

[lmedin02]

Ok, I have an idea on how to prove this now. Any finite cover of A must also cover the interval (0,1). Thus the sum of the lengths of the intervals that cover A must be greater than 1 since they also cover the interval (0,1). However, I still am having difficulty writing down an argument to show that any finite cover of A also covers the interval (0,1).