In the back of my textbook there is a proof that shows $ B = B^+$ iff $ BB \subseteq B$. There is a portion of the proof that is hard for me to follow:
Assume that $BB \subseteq B$ and show that $ B = B^+$ For every language $ B\subseteq B^+$ so we need only show $B^+ \subseteq B $. Let $w \in B^+ $, $w = x_1x_2x_3...x_k $ where $x_i \in B$ and $k \geq 1$. Because $x_1,x_2 \in B $ and $ BB \subseteq B $, we have that $x_1x_2 \in B$. Similarly, because $x_1x_2 \in B $ and $x_3 \in B$, we have $x_1x_2x_3 \in B$....
What confuses me is how does this not continue such that B is an infinite set? And if so, can we talk about infinite sets being subsets of other sets?