show that the function $f(x)=\frac{e^{-x}}{1+t^2}$ satisfies Lipschitz condition

Question

show that $$f(x)=\frac{e^{-x}}{1+t^2}$$ is defined $0<x<p, 0<t<N$ ( where N is positive integer ) Lipschitz condition with Lipschitz constant $N$

Attempt

consider $|f(t,x_1)-f(t,x_2)=|\frac{e^{-x_1}}{1+t^2}-\frac{e^{-x_2}}{1+t^2}|\\=|\frac{1}{1+t^2}(e^{-x_1}-e^{-x_2})|$

from here how to we procesed

Answer 1

$\frac 1 {1+t^{2}} \leq 1$ and $|e^{-x}-e^{-y}| \leq |x-y|$ for $x,y \geq 0$. The second inequality follows by MVT.