For a bivariate, real continuous function $f:\mathbb{R}^{2} \mapsto \mathbb{R}$, I need to show that $f(X_{n},Y_{n}) \to f(X,Y)$ in probability whenever $X_{n} \to X$ and $Y_{n}\to Y$ in probability. This is essentially a bivariate version of the Continuous Mapping Theorem for convergence in pro...

I am having trouble understanding gnometorule's answer to this question: Show that $[0,1]^{\omega}$ is not locally compact in the uniform topology, the uniform topology being induced by the metric $\displaystyle p(x,y) = \sup_{n \in \Bbb{N}} |x_n - y_n|$ Particularly, I am having trouble fo...

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $b ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $Ax = b$ has a solution in $\mathbb{R}^n$. Does it necessarily have a solution in $\mathbb{Q}^n$? and I thought I'd give an interesting, possibly wrong, approach t...