Every interval [x,y] is uncountable.

How to prove this? Thank you!

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- May 12th 2011, 03:50 PMjstarks44444Interval Countability proof
Every interval [x,y] is uncountable.

How to prove this? Thank you! - May 12th 2011, 04:16 PMPlato
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 $\displaystyle [a,b]\setminus \mathbb{Q}~?$ - May 12th 2011, 04:29 PMjstarks44444
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!