If $\ $ $ X\ =\ \left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}........\right\}$ will $ P(X)$ be countable?

Question

Question:

If $\ $ $ X\ =\ \left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}........\right\}$ will $ P(X)$ be countable?

My attempt:

I think $X$ is countable since you can list all the elements. But I am not sure if that implies $P(X)$ is countable.

Answer 1

$X$ is clearly countably infinite, so $P(X)$ has the same cardinality as $P(\mathbb N)$, which is uncountable by Cantor's theorem https://en.wikipedia.org/wiki/Cantor%27s_theorem.