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!

$$\zeta(4) < \sum_{n=1}^{6} \frac{1}{n^{4}} + \int_{6}^{\infty} \frac{dx}{x^4} < 1.1$$   
