This is exercise 6 from Chung's book: course in probability theory page 137.
If $(X_n)_n$ are independent and identically distributed with $E[X_1]=0$ and $(c_n)_n$ is a sequence of bounded real numbers, then $\frac{1}{n}\sum_{k=1}^n c_kX_k$ converges a.s to 0.
He put a hint : Truncate $X_n$ at $n$ and proceed as in the proof of the SLLN.
I succeeded solving the problem using his hint, but I have a question why to truncate $X_n$ at $n$ and not $c_nX_n$ at $n$? Also do we have the same result for the Marcinkiewics–Zygmund strong law of large numbers?