Example: Take the sequence . Extract three different subsequences of your choice and look at the convergence behavior of these subsequences.

Extracting the even terms yields the subsequence

• {1/2, 1/4, 1/6, 1/8, 1/10, ...}

Extracting the odd terms yields the subsequence

• {1, 1/3, 1/5, 1/7, 1/9, ...}

Extracting every third member yields the sequence

• {1, 1/4, 1/7, 1/10, 1/13, ...}

Hence, all three subsequences converge to zero. This is an illustration of a general result: if a sequence converges to a limit L then every subsequence extracted from it will also converge to that limit L.