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

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!