show $[0,1]$ is uncountable using outer measure

Question

This is a question from Real Analysis by Royden, 4th edition. (#5, pg. 34)

Using properties of outer measure, prove that $[0,1]$ is uncountable.

I believe that I am going to have to assume otherwise and use countable subadditivity in some way to produce a contradiction. I am not sure how to do so though. What other properties of outer measure might I need?

Thank you for any help!

Answer 1

Do you already know points have measure zero?

If so, then enumerate all points in $[0,1]$. The "countable" union of all these points is the entire interval, with measure $1$. The "countable" sum of the measures of all these points, on the other hand, is...

Answer 2

Let [0 1] is countable then outer measure of [0 1] is "zero" because outer measure of countable st is zero . and we also know that outer measure of [a b] is "b-a" so m*[0 1]=1-0=1. This is contradiction our assumption so [0 1] is uncountable.