How do I prove that an element or multiple elements are in both H and K?

Question

Let H, K $\leqslant$ Q (H and K are subgroups of Q(rational numbers)) (operation is addition). Suppose that H $\neq$ {e} and K $\neq$ {e}. Prove that H ∩ K $\neq$ {e}. So if I have an element in H, how do I prove that the element (and/or other elements) is also in K and thus also in their intersection?

Answer 1

Let $\frac{a}{b} \in H$ and $\frac{c}{d}\in K$. By the closure of addition, you have $a\in H$ and $c\in K$. So, $ac \in H\cap K$.