Question

Prove the set of real numbers is uncountable

Answer 1

these are the questions from the class that i've been skipping for a while now

Answer 2

Here's the proof I don't understand:
Assume real numbers is countable,
then we can rewrite it as a sequence:
{\[\mathbb{R} =\left\{ x_{1},x_{2},x_{3},...x_{n}... \right\}\]
Then every element of real numbers would be in that sequences.
Then we construct an interval:

So
\[I_{n+1} \subseteq I\]
\[x_{n} \notin I_{n}\]
Then if we pick any number from the list,
that number is not in an interval.
So as n approaches infinity,
\[x_{n} \notin \cap I_{n}\]
So the intersection of all intervals is empty.
But the Nested Interval Property states that the intersection of nested intervals for real numbers is NOT empty, so we have a contradiction.
This means that there exists a number in the intersection of all nested Intervals such that it's not in the sequence.
Therefore "an enumeration of R is impossible"
therefore R is uncountable.





I got lost at how the intervals represent the sequences of numbers in this proof. It's so confusing.