whether lipschitz continuous in one variable and continuous in another variable can imply continuity

I am considering a two-variable function\begin{align*}f:X\times Y&\to\mathbb{R}.\end{align*}Assume that $f(\cdot,y)$ is Lipschitz continuous for any $y\in Y$ and $f(x,\cdot)$ is continuous for any $x\in X$.
Now, we have two sequences $x_{n}\to x$ in $X$ and $y_{n}\to y$ in $Y$. Can we prove that$$f(x_{n},y_{n})\to f(x,y)$$My thought is that this is equivalent to prove that $f$ is continuous on $X\times Y$. So I was thinking for any fixed $(x,y)\in X\times Y$\begin{align*}|f(x,y)-f(x^{\prime},y^{\prime})|&\leq |f(x^{\prime},y^{\prime})-f(x,y^{\prime})|+|f(x,y)-f(x,y^{\prime})|\\&\leq L|x-x^{\prime}|+C(x)|y-y^{\prime}|\\&\leq C^{\prime}(x)\lVert (x,y)-(x^{\prime},y^{\prime})\rVert.\end{align*}This gives the continuity and then we are done. Does this proof makes any sense?