is $F_2$ a subgroup of any other $F_n$ for $n\geq 2$

Question

This is more to check an argument Since $F_2$ is a group generated by words of two generators, call them $\{a,b\}$ now every other $F_n$ (provided $n\geq 2$) will be all free words of more generators $\{a,b,c,d,...\}$ so it will surely contain words made up of only $a$ and $b$ and attaching one word with only letters $a$ and $b$ to another words containing $a$ and $b$ will still be a word containing only $a$ and $b$, this $F_2$ is closed under group multiplication. Is this argument correct?