Show that $O(n,\mathbb{R})$ the set of all orthogonal matrices is closed in $M(n,\mathbb{R})$ [duplicate]

You proof seems good to me.

Using matrix formalism can make it simpler: Let $\{M_p\}_{p\in\mathbb N}$ be the sequence of orthogonal matrices (so $M^T_pM_p=I$) and let $$M=\lim_{p\rightarrow+\infty}M_p$$ Then $$\begin{split} \|M^TM-I\| &= \|(M-M_p)^T M +M_p^T(M-M_p) +\underbrace{M_p^TM_p-I}_{=0}\|\\ &\leq \|(M-M_p)^T M\| + \|M_p^T(M-M_p)\| \\ &\leq \underbrace{\|(M-M_p)^T\|}_{\rightarrow 0}\| M\| + \|M_p^T\|\underbrace{\|(M-M_p)\|}_{\rightarrow 0} \\ \end{split}$$ Thus $M^TM=I$ and $M$ is orthogonal.

In general, when you can express a property (e.g. orthogonality) via a continuous function $f$, that is having the property is equivalent to $f(M_p)=0$, then limits preserve that property: $$0 = \lim_{p\rightarrow+\infty}f(M_p) = f(\lim_{p\rightarrow+\infty}M_p)$$

In your case, $f(M_p)=M_p^T M_p - I$.