$c_{00}$ represets all complex sequence with finitely many non-zero terms. $f$ is map from $c_{00} \mapsto \mathbb{C}$, I have to find the norm of $f$. Norm on $c_{00}$ is $\sum x_i\bar{y}_i$ My effort: If I consider the formula of sum of square, it's $n(n+1)(2n+1)/6$, and I was thinking maybe this $6$ is related to $\sqrt{6}$ of the question. Beyond this I don't know how to proceed.
Proof after comment: $\sum x_i/i < \sum 1/i$ because $\mid x \mid = 1$
Now $(\sum_{i}^{n} 1/i)^2 \leq \sum 1/i^2 = \pi^2/6$ because the cross term in the finite summation will be taken care by the term of form $1/i_1i_2$ in infinite sum. Now the result follows.