Consider simple random walk: $X_n=\pm1$ with equal probabilities. $S_n =\sum_{i=1}^nX_i$. For finite $n$ we can write
$$S_n=\sum_{i=1}^nX_i=\sum_{i=1}^nX_i^+ -\sum_{i=1}^nX_i^-$$ So that $$E[S_n]=\sum_{i=1}^nE[X_i^+] -\sum_{i=1}^nE[X_i^-]$$
However, $\lim_{n\rightarrow\infty}\sum_{i=1}^nE[X_i^+]=\infty$ and $\lim_{n\rightarrow\infty}\sum_{i=1}^nE[X_i^-]=\infty$.
So, what happens to $E[S_n]$ in the limit?
Simply we have $$E(S_n) = E(\sum_{i=1}^n X_i) =\sum_{i=1}^n E( X_i) = \sum_{i=1}^n 0 = 0$$ so $\lim_{n\to\infty}E(S_n) = \lim_{n\to\infty} 0 = 0$