If $A\subseteq B$, then is $\langle A \rangle \le \langle B \rangle$?

Question

For instance, if we have a group $\langle x_1, x_2 \rangle$, then is $\langle x_1 \rangle$ a subgroup of this? I would think that this is not true since every element in $\langle x_1, x_2 \rangle$ can be represented as $x_1^{\epsilon_1}x_2^{\epsilon_2}$ for $\epsilon_i \in \{-1, +1\}$ and $x_1^a \in \langle x_1 \rangle$ for any $a>1$ cannot necessarily be represented in this form. However, I am not sure if this is a valid argument.

Answer 1

Let $G$ be the group that you are working with. The group $\langle A\rangle$ is the intersection of all subgroups of $G$ containing $A$. Since $A\subset B$, $\langle B\rangle$ is one such subgroup, and therefore $\langle A\rangle$ is a subgroup of $\langle B\rangle$.