Can I assume the continum hypothesis in a proof I am showing that the cantor ternary set has the same cardinality as $\mathbb{R}$
I want to use the fact that it is uncountably infinite and a subset of $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$)
If I assume the continum hypothesis, the proof is done, but can I really do this? Is it a valid proof?
Notes:
The continum hypothesis:
There is no set whose cardinality is strictly between that of the integers and the real numbers.
Definition of Cantor Ternary Set: https://en.wikipedia.org/wiki/Cantor_set

You may prove it under the continuum hypothesis, but in your case CH in unnecessary. You can construct a explicit bijection $f$ between the Cantor set and $\mathcal{P}(\Bbb{N})$, namely
$$n\in f(x)\iff \text{$n$th digit of tenary expansion of $x$ after dot is 2}$$
(The tenary expansion of $x$ may not be unique, if $x$ is rational. In that case, choose the expansion which contains no 1. Such representation is unique.)
If possible, you should avoid using the continuum hypothesis. Our usual mathematics does not assume the continuum hypothesis and there are lot of (philosophical) disputes whether the continuum hypothesis is 'correct' axiom or not. 
If you prove something under continuum hypothesis, people who does not accept the continuum hypothesis (or even accept the axiom contradicted with the CH) would not accept your result.
