McShane's theorem states that if $(X,d)$ is a metric space, $S \subset X$ and $f: S \rightarrow \mathbb{R}$ is a $L$-Lipschitz function, then the functions $F,G : X \rightarrow \mathbb{R}$ defined for all $x \in X $ by $$F(x) =\sup_{y\in S}[f(y)-Ld(x,y )] \hspace{0.3cm} \text{y} \hspace{0.3cm} G(x) = \inf_{y' \in S}[f(y') + Ld(x,y')]$$ are $L$-Lipschitz extensions of $f$ to all $X$
Now, I know that I can apply this theorem to $L$-Lipschitz functions with values in $\mathbb{R}^n$ to obtain an extension of $f$ with Lipschitz constant greater than $L$ for $n>1 $. I don't understand how to apply the above theorem to each component. So, I would like to see a particular example when I have a function $L$-Lipschitz $f:S \rightarrow \mathbb{R}^2$, or if I have $\mathbb{C}$ instead of $\mathbb{ R}^2$ . How do I apply McShane's theorem to get a Lipschitz extension for $f$? I know that the extension will not be $L$-Lipschitz, in fact, my purpose is to verify that the Lipschitz extension will not have the same Lipschitz constant when applying McSchane's theorem for Euclidean spaces of dimension $n>1$.
Given $f:S\to\mathbb{R}^n$, write it down as $f=(f_1,\ldots,f_n)$, $f_i:S\to\mathbb{R}$. If $f$ is $L$-Lipschitz then
$$|f_i(x)-f_i(y)|=\sqrt{(f_i(x)-f_i(y))^2}\leq $$ $$\leq\sqrt{\sum_{j=1}^n(f_j(x)-f_j(y))^2}=\lVert f(x)-f(y)\rVert\leq Ld(x,y)$$
meaning each $f_i$ is $L$-Lipschitz as well. Therefore, by McShane's theorem, each $f_i$ has a $L$-Lipschitz extension $F_i$ to whole $X$. Then we put $F=(F_1,\ldots,F_n)$. Clearly $F$ is an extension of $f$, and we have
$$\lVert F(x)-F(y)\rVert=\sqrt{\sum_{j=1}^n(F_j(x)-F_j(y))^2}\leq$$ $$\leq\sum_{j=1}^n\sqrt{(F_j(x)-F_j(y))^2}=\sum_{j=1}^n | F_j(x)-F_j(y)|\leq$$ $$\leq \sum_{j=1}^n Ld(x,y)=nLd(x,y)$$
meaning $F$ is $nL$-Lipschitz extension of $L$-Lipschitz function $f$.
EDIT: As mentioned in comments, in fact we can do better. Since $|F_j(x)-F_j(y)|\leq Ld(x,y)$ then
$$\lVert F(x)-F(y)\rVert=\sqrt{\sum_{j=1}^n(F_j(x)-F_j(y))^2}\leq$$ $$\leq\sqrt{\sum_{j=1}^n(Ld(x,y))^2}=\sqrt{n(Ld(x,y))^2}=$$ $$=\sqrt{n}Ld(x,y)$$
meaning $F$ is in fact $\sqrt{n}L$-Lipschitz.