We know that the hyperbolic tangent function, $\tanh x$, is less than one. I want to show that it is also less than a function which can be smaller than one. In particular, I want to prove that \begin{align}\tanh \frac\pi 2x\le\frac\pi2\frac x{\sqrt{1+x^2}}.\end{align} If $\frac x{\sqrt{1+x^2}}\ge\frac2\pi$, then $\frac\pi2\frac x{\sqrt{1+x^2}}\ge\frac\pi2(\frac2\pi)=1\ge\tanh \frac\pi 2x$ and the inequality holds. My question is: How can i show that this inequality is also true for the case $\frac x{\sqrt{1+x^2}}<\frac2\pi$ ?.
I am working on something, and if i show this last case then I will be done. Thanking you in advance.