Here's a homework question I'm trying to solve:
Prove or disprove: if $\lim_af$ and $\lim_ag$ do not exist, then $\lim_a(f \cdot g)$ do not exist either.
So I know that $(\forall l\in\mathbb{R})(\exists\epsilon\gt0)(\forall\delta_1\gt0):(\|x-a\|\lt\delta_1)(\rightarrow\|f(x)-l\|\ge\epsilon/2)$ $(\forall m\in\mathbb{R})(\exists\epsilon\gt0)(\forall\delta_2\gt0):(\|x-a\|\lt\delta_2)(\rightarrow\|g(x)-m\|\ge\epsilon/2)$
Now, since this is true for every $l,m\in\mathbb{R}$, it's also true for for every $r\in\mathbb{R}, r=m\cdot n$. In the same way, the two statements hold for every $\delta\gt0$ then $(\forall r\in\mathbb{R})(\exists\epsilon\gt0)(\forall\delta\gt0):(\|x-a\|\lt\delta)(\rightarrow\|f(x)-l\| \cdot \|g(x)-m\|\ge\epsilon/2 \cdot \epsilon/2)$
How do I continue from here, assuming I was right so far?
Thanks