Hy, I hope everyone is doing well. I just recalled a problem that have been bothering me. I have already posted it here Challenging probability problem but it seems that I still can not figure out how to complete the solution formalized by DLeMeur, he used the fact that $\frac{s_{n}}{\sqrt{n}}$ has limit sup $1$ and limit inf $-1$, which is incorrect actually it is $+\infty$ and $-\infty$ so the argument seems to fail.
Is it possible to obtain the estimate $|s_{n}| \leq C(u, \varepsilon) + (1+\varepsilon) \sqrt{n}$ by an other way ?
If we manage to do so DLeMeur's idea can still work to solve the problem.
Edit: As pointed out this estimate is false, Any idea to solve the problem ?