In the book "An Introduction to Mathematical Analysis for Economic Theory and Econometrics" by Dean Corbae, I saw the proof for Tietze extension theorem, which states that "If $M$ is a metric space , $F$ is a closed set in $M$ and $g \in C_b(F)$ , then there is a continuous extension $G \in C_b(M)$ such that $\|G \|_{\infty}=\|g \|_{\infty}.$"
They consider the following extension:
$$G(x)=\begin{cases} g(x) &\text{ if } x \in F \\ \sup_{t \in F}\dfrac{g(t)}{(1+d(t,x)^2)^{1/d(x,F)}}& \text{ if } x\notin F \end{cases}$$
For proving $G(x)$ is continuous at $x \in \partial F$, they consider a sequence $x_n \subset F^c$ which converges to $x$. For $t \neq x$, $\dfrac{1}{(1+d(t,x_n)^2)^{1/d(x_n,F)}} \overset{n \to \infty}{\longrightarrow} 0$ and $\dfrac{1}{(1+d(x,x_n)^2)^{1/d(x_n,F)}} \overset{n \to \infty}{\longrightarrow} 1$. They mention to use the property: $\lim_{s \to 0}f(s)=1$, where $f(s)$ is strictly decreasing function $$f:(0,+\infty) \to (0,1): s \mapsto f(s)=\dfrac{1}{(1+s^2)^{1/s}}$$ Then $G(x_n)\overset{n \to \infty}{\longrightarrow} G(x)$.
My question is that how they use above property of $f(s)$ to prove $a_n = \dfrac{1}{(1+d(x,x_n)^2)^{1/d(x_n,F)}} \overset{n \to \infty}{\longrightarrow} 1$. I know that both two terms $d(x,x_n)$ and $d(x_n,F)$ converge to $0$ but they are not the same "$s$" as in function $f(s)$. Any other way to prove continuity of $G(x)$ is good too. Thanks in advance.
Note that $x\in F$, and that therefore $d(x_n,F)\leqslant d(x_n,x)$. So,$$f\bigl(d(x_n,x)\bigr)=\frac1{(1+d(x_n,x)^2)^{1/d(x_n,x)}}\leqslant\frac1{(1+d(x,x_n)^2)^{1/d(x_n,F)}}\leqslant1.$$Now, apply the squeeze theorem.