Every interval [x,y] is uncountable.
How to prove this? Thank you!
Without knowing what theorems go before this question, there is no way to give you an answer. For example: if $\displaystyle a<b$ then the interval $\displaystyle [a,b]$ is uncountable.
The rational numbers are countable.
So what does mean about $\displaystyle [a,b]\setminus \mathbb{Q}~?$
Ah, you're right. The theorems that come before this include,
R (Reals) is not countable.
The set R - Q (Rationals) of irrational numbers is uncountable.
The set of transcendental numbers is uncountable.
The proof of the reals not being countable shows that the set of decimals
{0.d1d2d3...: each dj = 3 or 4} which is a subset of R
is uncountable. Consequently, the interval [0,1] = {x in R : 0 <= x <= 1} is uncountable. This can apparently be modified to prove the theorem in question. Thanks!