Prove the following inequality $x^2+y^2+1>x\sqrt{y^2+1}+y\sqrt{x^2+1}$ Can anybody help me prove this inequality ?

\begin{align*}
x\sqrt{y^{2}+1}+y\sqrt{x^{2}+1}&=\left(x,y\right)\cdot\left(\sqrt{y^{2}+1},\sqrt{x^{2}+1}\right)\le\sqrt{(x^{2}+y^{2})(x^{2}+y^{2}+2)}\\
&\le\frac{2x^{2}+2y^{2}+2}{2}=x^{2}+y^{2}+1
\end{align*}
where I have used Cauchy-Schwarz followed by the arithmetic-geometric inequality. Note that equality cannot occur due to the equality case of the arithmetic geometric inequality.
