Prove that $\zeta (4)\le 1.1$ Prove the following inequality
$$\zeta (4)\le 1.1$$
I saw on the site some proofs for $\zeta(4)$ that use Fourier or Euler's way for computing its precise value, and that's fine and I can use it. Still, I wonder if there is a simpler  way around for proving
this inequality.  Thanks!

$$
\begin{eqnarray}
\zeta(4)&<&\sum_{n=1}^{9}\frac{1}{n^4} +\left(\sum_{n=10}^{\infty}\frac{1}{n^2}\right)^2 \\
&=& \sum_{n=1}^{9}\frac{1}{n^4} + \left(\frac{\pi^2}{6}-\sum_{n=1}^{9}\frac{1}{n^2}\right)^2 \\
&=& 1.0929965...
\end{eqnarray}
$$
