Proving that the sequence $a_{n}= \frac{1}{n}$ ($n$ odd) and $\frac{3}{n}$ ($n$ even) converges.
Consider the sequence $b_n=\frac{3}{n}$ then $a_n\leq b_n$ for every $n\in \mathbb{N}$. It is easy to see that $b_n$ converges, now use this to show that $a_n$ converges too.