Prove, that $A^+\subseteq A^*$ where $A$ is a formal language
It seems you have hit the reasoning correctly:
A set $S$ is a subset of another set $A$ if $w\in S\implies w\in A$.
Since you are trying to show that $\forall w \in A^+ \implies w \in A^*$, you are proving that $ A^+ \subseteq A^*$.

One more legant way to formally do this is by finding a bijection between the two sets.
Since $|A^*|$ = $|\Bbb N|$ you need to prove that $|A^+|$ is a subset of  $|\Bbb N|$. So you need to find a Bijection (left to you since we dont do Homework for the OP's) from these two sets.
Note that there exists $2^{\aleph_0}$ such functions, so 
 you have a large selection space :)