Random Walk With Fair Coin Finding the limit

Question

Let us $S_n = X_1 +\cdots +X_n $ where $X_i$ is 1 if the outcome is heads and $X_i$ is -1 if the outcome is tail with a fair coin flip of n time. Find the $\lim_{n \to \infty} \frac{S_n}{n}$. Ok. The answer is given as Zero. but I can't convince myself to access by understanding it. My thinking is that: We first find the $\lim_{n \to \infty} S_n$, which is undefined (if I still have some correct understanding of limit.) How come the limit of $\lim_{n \to \infty} \frac{S_n}{n}$ is zero since the limit of $\lim_{n \to \infty} S_n$ is already undefined? Any hints or different perceptive to look at it?

Answer 1

Probability of heads is 1/2 and same for tails. So for a large number of trials there should be equal number of heads and tails thus the Sn becomes zero that is why limit is zero.