We know that $S = \sum _{n=1}^{n=\infty} \frac{1}{n^2} = \frac{\pi^2}{6} = 1.64493406682264364..$. By the definition of convergence, there exists some $N_0 \in \mathbb{N}$ such that $S_n > 1.64493406682264363$ for all $n > N_0$ where $S_n$ denotes the partial sum. Is there a way to find such $N_0$ analytically? If not, have we really proved Basel's problem?
PS : Please note here that a computer generated proof is not allowed. I'm not looking so much for a traditional proof, but to find such $N_0$. The spirit of the question is whether it is possible to find such $N_0$ for any given number less than $\frac{\pi ^2}{6}$ without using computers to evaluate the actual series.
Use Cauchy Integral Test - Bounds.
For your information, the famous Basel problem is solved.