I'm working through some textbook problems and ran into this. I've seen a few solutions that reduce the problem to multiplication but the textbook didn't allow me that. I'm still inexperienced with these sort of proofs and am curious if I'm lacking any rigour.
I want to prove that $$\forall\varepsilon>0\exists N>0|n>N\rightarrow\left|\frac{a_{n}}{b_{n}}-\frac{L}{M}\right|<\varepsilon$$, given that $a_n$ coverges to $L$ and $b_n$ converges to $M\neq0$. I have proven it as follows,
Suppose,
$$\forall\varepsilon>0\exists N_{1}>0|n>N_{1}\rightarrow\left|a_{n}-L\right|<\frac{\varepsilon}{2\left|M\right|}$$
$$\forall\varepsilon>0\exists N_{2}>0|n>N_{2}\rightarrow\left|b_{n}-M\right|<\frac{\varepsilon}{2\left|L\right|},M\neq0$$
Let $\varepsilon>0$ be given
Choose $N=\max\{N_{1},N_{2}\}>0$
If $n>N$,
Then $\left|a_{n}-l\right|=\left|\frac{a_{n}}{b_{n}}-\frac{L}{M}\right|$
$=\frac{\left|Ma_{n}-Lb_{n}\right|}{\left|Mb_{n}\right|} $Property of absolute value
$=\frac{\left|Ma_{n}-ML-Lb_{n}+ML\right|}{\left|Mb_{n}\right|}$
$\leq\frac{\left|Ma_{n}-ML\right|+\left|ML-Lb_{n}\right|}{\left|Mb_{n}\right|} $Triangle inequality
$=\frac{\left|M\right|\left|a_{n}-L\right|+\left|L\right|\left|M-b_{n}\right|}{\left|M\right|\left|b_{n}\right|} $Property of absolute value
$<\frac{\varepsilon}{\left|M\right|\left|b_{n}\right|} $ By the supposed information
$<\varepsilon$ Since the denominators are positive and greater than 0