Proving that $|\{B\subseteq S: |B|<\infty \}|=|S|$

Question

I've some elementary set theory problem that I came across with:

Let $S\subseteq\mathbb{R}$ be infinite set, and let $A=\{B\subseteq S: |B|<\infty \}$. I'm interested in showing that cardinality of set $A$ is equals to cardinality of $S$, i.e $|A|=|S|$.

I thought using Cantor–Bernstein–Schroeder theorem in some way.

Answer 1

Hint What is the cardinality of the set of singleton subsets of $S$?

What is the cardinality of the set of two-element subsets of $S$?

$\cdots$

What is the cardinality of the set of $n$-element subsets of $S$?

$\cdots$