Let $X_{n,k}$ be a double sequence of random variables.
Assume for each fixed $k$, $X_{n,k}\xrightarrow[n\rightarrow\infty]{p}\alpha_k$, where $\alpha_k$ is a non-random scalar. Assume further that $\alpha_k\xrightarrow[k\rightarrow\infty]{p.w} \alpha$, where $\alpha$ is again non-random. Is it possible to show the following?
$$X_{n,k}\xrightarrow[n,k\rightarrow\infty]{p} \alpha$$
My attempt: Let $f$ be any arbitrary Lipschitz Cont. function bounded by $M<\infty$ and satisfying the contraction $\vert f(x_1) - f(x_2)\vert \leq K\vert x_1 - x_2\vert$ for some $K$ (by definition). Then consider \begin{align} \mathbb{E}\big\vert f(X_{n,k}) - f(\alpha)\big\vert&\leq\mathbb{E}\big[\big\vert f(X_{n,k}) - f(\alpha_k)\big\vert\big] + \big\vert f(\alpha_k) - f(\alpha)\big\vert\\ &=\mathbb{E}\big\vert f(X_{n,k}) - f(\alpha_k)\big\vert\mathbb{1}_{(\vert X_{n,k} - \alpha_k\vert\leq \epsilon)} + \mathbb{E}\big\vert f(X_{n,k}) - f(\alpha_k)\big\vert\mathbb{1}_{(\vert X_{n,k} - \alpha_k\vert> \epsilon)} + \big\vert f(\alpha_k) - f(\alpha)\big\vert\\ &\leq K\epsilon + M\mathbb{P}[\vert X_{n,k} - \alpha_k\vert> \epsilon] + K\vert \alpha_k - \alpha\vert \end{align}
Now, we know that there exists a $K^*$ large enough such $K\vert \alpha_k - \alpha\vert\leq K\delta$ for all $\delta>0$
Similarly, $\mathbb{P}[\vert X_{n,k} - \alpha_k\vert> \epsilon]<\delta$ for large enough $N^{*}$.
Is it enough to choose $N = \max(N^{*},K^{*})$ to show that $\mathbb{E}\big\vert f(X_{n,k}) - f(\alpha)\big\vert\leq \delta$ for all $n,k\geq N$? Then I could use the Portmanteau lemma to show the rest.