Finding lipschitz bound for $F(x_1,\ldots,x_n) = (\tanh(x_1),\dots,\tanh(x_n))$

Question

Given a vector-valued function $F(x) = (f_1(x),\ldots,f_m(x))$, where $x = (x_1,\ldots,x_n)$. Taking for example uniformly $f_j(x)=\tanh(x)$, how can I prove that $\lVert F(x) - F(y)\rVert_2 \leq\lVert x-y\rVert_2$ for vectors $x,y$? This is a fundamental step in the proof of Echo State Property, the stability theorem of Echo State Network (see at pag 41 of https://pdfs.semanticscholar.org/8430/c0b9afa478ae660398704b11dca1221ccf22.pdf, Proof of Proposition 3).

Answer 1

I got the answer using Mean Value Theorem for scalar-value tanh, from which I have $|tanh(x)-tanh(y)|\leq|x-y|$ per every $x,y \in \mathbb{R}$; now setting the norm I use this result to limit $||F(x)-F(y)||$ where $x,y \in \mathbb{R}^n$ through trivial algebra.