1.2 Subsequences
If we have a sequence \((a_n)_{n}\), we can just as easily make a new sequence by selecting members of \((a_n)_{n}\). These will turn out to be useful when trying to prove results about functions in a few weeks.
As a (slightly boring) example, suppose that the terms of a sequence \((a_n)_n\) are given by \(a_n = n\). Then, possible choices for the sequence \((n_k)_k\) are
- \(n_k = 2k\). This choice gives us the subsequence of \((a_n)_n\) containing all the even numbers.
- \(n_k = k^{\text{th}}\, \text{prime}\). This choice gives us the subsequence of \((a_n)_n\) consisting of all prime numbers. Note that we may not know each possible \(n_k\), but \((n_k)_k\) is still a strictly increasing sequence.
Moreover, since \((n_k)_{k\in\mathbb{N}}\) is a strictly increasing sequence, we must have that \(n_k \geq k\). This is because for each \(k \in \mathbb{N}\), the first possible choice we could make for \(a_{n_k}\) in our subsequence is simply \(a_k\).
We also have a few useful results regarding subsequences:If a real sequence \((a_n)_n\) converges to a limit \(L\), then all subsequences \((a_{n_k})_k\) of \((a_n)_n\) also converge to \(L\).
The contrapositive of this result is very good for proving that sequences do not converge, i.e. if you can find two subsequences converging to different limits, then the original sequence does not converge. We also have the following result linking bounded sequences and subsequences.
Every bounded real sequence has a convergent subsequence.
Following the explanation here, this theorem says that no matter how ‘random’ a sequence might be, as long as its bounded then some part of it must converge. This theorem may seem innocuous for now, but you’ll see it appear in later courses to prove some key results in analysis.