Q: $F(a,b)$ is free group of rank 2, give an example (and proof it!) of free subgroup with infinity rank.
My attempt: I was trying with $G=[F(a,b), F(a,b)]$ and $G=\langle {a^nb^n : n\in N}\rangle $
Both of these are subgroups of $F(a,b)$ (i can use the fact that first one is normal subgroup of $F(a,b)$, and second one is a group from it's definition). Now I have a problem with showing that ${[a^n,b^m] : n,m\in N}$ is a set of free generators in first example, and $a^nb^n : n\in N$ in second example. I saw many proofs of the first one but these were using too advanced stuff, I am new in group theory, don't know graphs and algebraic topology.