Are there countably or uncountably many infinite subsets of the positive even integers? Let $S$ be the set of all infinite subsets of $\mathbb N$ such that $S$ consists only of even numbers.
Is $S$ countable or uncountable?
I know that set $F$ of all finite subsets of $\mathbb N$ is countable but from that I am not able to deduce that $S$ is uncountable since it looks hard to find a bijection between $S$ and $P(\mathbb N)\setminus F$. Also I am not finding the way at the moment to find any bijection between $S$ and $[0,1]$ to show that $S$ is uncountable nor I can find any bijection between $S$ and $\mathbb N$ or $S$ and $\mathbb Q$ to show that it is countable. So I am thinking is there some clever way to show what is the cardinality of $S$ by avoiding bijectivity arguments?
So can you help me?

Use the Cantor-Bernstein theorem, that is (in the simplified version)
$$|A| \leq |B| \text{ and } |B| \leq |A| \text{ implies } |A| = |B|.$$
We have that $S \subset P(\mathbb{N})$, so $|S| \leq |P(\mathbb{N})|$. Moreover, we have an injection $f : P(\mathbb{N}) \to S$ given by 
$$ f(A) = \{4k \mid k \in A\} \cup \{4k+2 \mid k \in \mathbb{N}\}. $$
Hence, we have $|P(\mathbb{N})| \leq |S|$, so $|S| = |P(\mathbb{N})|$.
I hope this helps ;-)
