I'm self-studying from Algebra: Chapter 0 by Aluffi. I was working on Exercise 6.6, and I can do the first part (which I've included for context), but I'm having trouble with the second part.
Here is Exercise 6.6:
Prove that the union of a family of subgroups of a group G is not necessarily a subgroup of G. In fact:
Let $H$, $H'$ be subgroups of a group $G$. Prove that $H \cup H'$ is a subgroup of G only if $H \subseteq H'$ or $H' \subseteq H$.
On the other hand, let $H_0 \subseteq H_1 \subseteq H_2 \subseteq ...$ be subgroups of a group $G$. Prove that $\cup_{i\geq0}H_i$ is a subgroup of $G$.
The second part is easy in the case that the sequence of subgroups is finite. Say that there is a last subgroup in the sequence - call it $H_{last}$. Then $\cup_{i\geq0}H_i$ is just $H_{last}$, which we know is a subgroup of G.
However, I'm really not sure how to approach the second part if the sequence is infinite. Do we have to do something analogous to taking a limit in calculus?
(I'm an engineering major and haven't taken any analysis or abstract algebra classes, so if you could keep that in mind when answering that would be great!)