Sequences · Definition

Subsequence

real analysissequences

Before you start you need to know

It lets us study a sequence by looking at only some of its terms, and it is the tool to prove that a sequence does not converge or that its terms cluster around certain values.

Definition

A subsequence of a sequence \((a_n)\) is a sequence \((a_{n_k})\) made of terms of \((a_n)\) with strictly increasing natural indices \(n_k\).

\[a_{n_1},a_{n_2},a_{n_3},\dots\ \text{with}\ n_1

A subsequence is obtained by «skipping» terms without changing the order. If \(a_n=1/n\), the terms \(a_{2k}=1/(2k)\) form a subsequence; but \(1/2,1,1/4,1/3,\dots\) does not, because it changes the order. Since the indices increase strictly, \(n_k\ge k\) for all \(k\), so \(n_k\to\infty\).

The basic result is that if \(a_n\to L\), every subsequence also converges to \(L\): given \(\varepsilon>0\) there is \(N\) with \(|a_n-L|<\varepsilon\) for \(n\ge N\), and since \(n_k\ge k\), it suffices to take \(k\ge N\). The contrapositive proves divergence: if two subsequences have different limits, or one subsequence diverges, the original sequence does not converge.

The classic example is \(a_n=(-1)^n\): the even-index subsequence is constantly \(1\) and the odd-index one is constantly \(-1\), so it does not converge. Note that the converse fails: a sequence may have convergent subsequences without being convergent.

This concept prepares the Bolzano-Weierstrass theorem, which says every bounded sequence has at least one convergent subsequence, and later compactness, where that property becomes a definition.

Think about it

If one subsequence of \((a_n)\) converges to \(3\) and another to \(5\), what follows about \((a_n)\)?

It diverges: if it converged to \(L\), all its subsequences would have limit \(L\).

If \((a_{2k})\) and \((a_{2k+1})\) converge to the same limit \(L\), does \((a_n)\) converge?

Yes, to \(L\): given \(\varepsilon\), one \(N\) works for the even terms and another for the odd ones; the larger works for every \(n\).

Exercises 10

  1. Let \(a_n=n^2\) and \(n_k=2k\). What is the third term \(a_{n_3}\) of the subsequence?
  2. Which of these sequences is a subsequence of \(1,\tfrac12,\tfrac13,\tfrac14,\dots\)?
  3. Let \(a_n=\frac{n+1}{n}\). To what number does the subsequence \(a_{k^2}\) converge?
  4. Let \(a_n=(-1)^n\). To what does the even-index subsequence \(a_{2k}\) converge?
  5. Let \(a_n=3+\frac1n\). To what does the subsequence \(a_{10^k}\) converge?
  6. Let \(a_n=\sin\left(\frac{n\pi}{2}\right)\). Which of these numbers is NOT the limit of any subsequence of \((a_n)\)?
  7. If \(a_{2k}\to1\) and \(a_{2k+1}\to2\), what happens to \((a_n)\)?
  8. Which of these index choices defines a subsequence \((a_{n_k})\) of any sequence \((a_n)\)?
  9. If \(n_1<n_2<\cdots\) are natural numbers, which statement always holds?
  10. Let \(a_n=(-1)^n\left(1+\frac1n\right)\). What is the set of limits of its convergent subsequences?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §2.5, Definición 2.5.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 3, Definición 3.5
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §3.4, Definición 3.4.1