Doubt about how to find a Lipschitz constant I have a doubt about a sentence of my Calculus text. 

Let $f: [t_1, t_2]\times \mathbb{R}^n \to \mathbb{R}^n, (t,y)\to f(t,y)$ such that  $|\partial_{y_j} f_i|$ is continuous and bounded for every $i,j=1,...n$. Then f has $\sqrt n L$ as Lipschitz constant (with respect to y uniformly in t) where $L>0$ is such that $|\partial_{y_j} f_i|\le L$.

I don't know how to get $\sqrt n L$ as Lipschitz constant: for $i=1,...,n$ we have $|f_i(t,y)-f_i(t,z)|\le|\nabla f_i(t, \theta y)||y-z|$ for some $\theta\in[0,1]$; and since $|\nabla f_i(t, \theta y)|\le \sqrt{nL^2} $ we obtain $|\nabla f(t, \theta y)|\le \sqrt{n^2L^2}=nL$.
Do you know ways to improve my inequality? Thanks in advance.

Indeed, I think that there is an error in the book. That the example of $n=2$ and $f$ the linear map with all coefficients equal to $1$. Then the $\partial_j f_i$ are also all equal to one. Take $x=(0,0)$ and $y=(1,1)$. You have $\Vert f(x)-f(y) \Vert_2 = 2\sqrt{2} =2 \Vert x-y\Vert_2 $.
