The preimage of continuous function on a closed set is closed. My proof is very different from my reference, hence I am wondering is I got this right?
Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of continuous function on closed set is closed.
Let $D$ be a closed set, Consider a sequence $x_n \to x_0$ in which $x_n \in f^{-1}(D)$, and we will show that $x_0 \in f^{-1}(D)$.
Since $f$ is continuous, we have a convergent sequence
$$\lim_{n\to \infty} f(x_n) = f(x_0) = y.$$
But we know $y$ is in the range, hence, $x_0$ is in the domain. So the preimage is also closed since it contains all the limit points.
Thank you.

Let me add an analogous approach that works for topological spaces. We need to introduce an additional concept that is basically a generalisation of sequences for topological spaces. 
Let $S$ be a directed set (reflexive, transitive and any pair of elements has a upper bound) and let $X$ be a topological space. Then a net is a function $x: S \rightarrow X$. We often use the notation $x_{\alpha}$ instead of $x(\alpha)$ for evaluation to mirror the notation used for sequences. And we say that a net converges to a point $x \in X$ when for all neighbourhoods of $x$ $N_{x}$ there exists an $\alpha_{0}$ such that
$$\left \{ x_{\alpha}:\alpha \geq \alpha_{0} \right \} \subset N_{x}.$$
We denote this fact by $x_{\alpha} \rightarrow x$.
Using this we can characterise closed set for general topological spaces in a "sequential manner". Let's compare the two characterisations that may be used:


*

*In a first countable space (every metric space is first countable), $A$ is closed iff for every sequence $(x_{n}) \subset A:x_{n} \rightarrow x$ we have $x \in A$.

*In a topological space (no additional separation assumptions are needed), $A$ is closed iff for every net $(x_{\alpha}) \subset A:x_{\alpha} \rightarrow x$ we have $x \in A$.


Note that we also have a theorem for nets that states that we can interchange a continuous function and the limiting operation. Thus allowing us to do exactly (notation wise) what 1LiterTears did in the initial post (consider the same setup, just replace sequence with net):
$$\text{lim}f(x_{\alpha}) = f(\text{lim}x_{\alpha})=f(x).$$
This of course allows us to also get the "same result" if we replace sequence by net in the initial statement.
In short, nets are a great way to move from first countable spaces to general topological spaces while still being able to recycle some of the "standard arguments" from analysis.
