The fact that all countable subsets of the real numbers have Lebesgue measure zero (as proved in the video) implies that the real numbers are uncountable. Other

The fact that all countable subsets of the real numbers have Lebesgue measure zero (as proved in the video) implies that the real numbers are uncountable. Otherwise the real numbers would have Lebesgue measure zero, which is absurd. This approach to proving R uncountable is carried out more formally here (proof A.3): https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-5%2F1.pdf https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-...