Given the sequence $(x_n)$ is unbounded, show that there exist a subsequence $(x_{n_k})$ such that $\lim(1/x_n)=0$.
Solution 1:
Hint: If $(x_n)$ is unbounded, then for each $N$ there are infinitely many $n$ with $|x_n| \ge N$. Now choose inductively $n_k$ such that $n_1 < n_2 < \cdots$ and $|x_{n_k}| > k$.