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

Extracting the even terms yields the subsequence

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

Extracting the odd terms yields the subsequence

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

Extracting every third member yields the sequence

{1, 1/4, 1/7, 1/10, 1/13, ...}
which converges to zero (prove it !).

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.

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