## Examples 3.3.2(b):

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

The sequence in question is:
which converges to zero. Now let us extract some subsequences:= {1, 1/2, 1/3, 1/4, 1/5, 1/6, ... }

Extracting the even terms yields the subsequence

which converges to zero (prove it !).{1/2, 1/4, 1/6, 1/8, 1/10, ...}

Extracting the odd terms yields the subsequence

which converges to zero (prove it !).{1, 1/3, 1/5, 1/7, 1/9, ...}

Extracting every third member yields the sequence

which converges to zero (prove it !).{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*.