Prove that if $X$ and $Y$ are sequences such that $X$ converges to $x\neq 0$ and $XY$ converges, then $Y$ converges

I want to prove that given convergent sequences $X$ and $XY$ then $Y$ converges.

Call arbitrarily $z_n=(x_n)(y_n)$, as by hypothesis $\lim(x_n)=x≠0$, then it must happen that $(x_n)≠0$, thus we can consider that $y_n=\frac{z_n}{x_n}$; now, assuming that $(z_n)$ tends to $z=xy$ and that $(x_n)$ converges to $x$ one has that $y_n$ converges to $y$.

The above represents an attempt at a proof, but I don't know if this is the way, any help?

What you want to show is

$$p_n\rightarrow p,q_n\rightarrow q\neq 0\implies p_n/q_n\implies p/q$$

To show this, first show the limit of a product of sequences is the product of limits (left as an exercise):

$$a_n\rightarrow a,b_n\rightarrow b\implies a_nb_n\rightarrow ab,$$

then show the limit of a reciprocal of a sequence is the reciprocal of the limit when well-defined (proven here), and then take the case $a_n=p_n,b_n=1/q_n.$