Random Walk and strong law

Question

I want to prove that a Random Walk in 1 dimension is transient when $p\neq\frac{1}{2}$ but i want to prove it by the strong law of large numbers, so i have this: Define a random variable $$X_i = \left\{ \begin{array}{l l} 1 & \quad \text{if you go to the right }\\ -1 & \quad \text{if you go to the left} \end{array} \right.$$ With probabilities $P(X_i=1)=p \quad, P(X_i=-1)=1-p$, where each $X_i$ is independent with the same distribution. Now by the strong law of large numbers $$\sum\limits_{i=1}^n\frac{X_i}{n}\to E[X_i] \quad as \quad n\to \infty $$ I want to prove that $\lim_{n \to +\infty}\sum\limits_{i=1}^nX_i=\infty$ when $p>\frac12$ i.e the position or state at time n is $\infty$, it means that the walk never returns to any state, hence the random walk is transient. But i don't know how to prove the last limit, and i don't understand my teacher's hint of use the strong law because i think is not necessary. Hope you help me.