Intro to Subsequences | Real Analysis

What are subsequences in real analysis? In today's lesson we'll define subsequences, and see examples and nonexamples of subsequences. We can learn a lot about a sequence by studying its subsequence, so let's talk about it!

If (a_n) is a sequence, we can denote a subsequence of (a_n) as (a_{n_k}). Here, {n_k} is itself an increasing sequence of positive integers, and you can think of it as a way of picking out the terms of (a_n) that will make up our subsequence. For example, suppose n_k = 2k-1, so that n_1 = 1, n_2 = 3, n_3 = 5, and so on. Then the subsequence (a_{n_k}) consists of the n_1th term of a_n, the n_2th term, the n_3th term, and so on. In this case, that means (a_{n_k}) consists of the n_1 = 1st term of (a_n), then the n_2 = 3rd term of (a_n), then the n_2 = 5th term of (a_n), and so on. The n_k values are picking out the terms of (a_n) to make up our subsequence.

This may sound complicated, but the idea is very simple. A subsequence of (a_n) is a sequence of terms, all from (a_n), where each term in the subsequence is further along in the original sequence. This means, for example, (a_2, a_3, a_5, a_7, a_11, ... ) could be a subsequence of (a_n), but (a_2, a_3, a_3, a_1, a_5, ...) is not, because this repeats a_3, and it goes from a_3 to a_1, which is moving backwards in the original sequence, which isn't allowed for a subsequence. As a result of how subsequences are defined, the kth term of a subsequence is at least k terms along in the original sequence.

Also, every sequence is a subsequence of itself.

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Intro to Sequences:    • Intro to Sequences | Calculus, Real A...  

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