How to prove/disprove this inequality?

Question

Given a series of independent random variables $\{X_i\}$, such that: $P(X_i= i^{1/2})=\frac{1}{2i^{1/2}}$ , $P(X_i=-(i^{1/2}))=\frac{1}{2i^{1/2}}$, $P(X_i=0)=1-\frac{1}{i^{1/2}}$.i is a natural number. Prove or disprove: limit of $P\left(\left|\frac{X_1+X_2...+X_n}{2} - \frac{1}{2}\right|< \frac{1}{100}\right)$ as $n\to \infty$ equals to $1$. It seems I could use chebychev or the weak law of large numbers somehow (I already managed to see that they all have the same expectation that equals zero) ...any ideas ?