Sequences · Definition

Bolzano-Weierstrass theorem

real analysissequences

Before you start you need to know

It guarantees limits without knowing the sequence: being bounded is enough for something inside it to converge.

Definition

Every bounded real sequence \((a_n)\) has at least one subsequence \((a_{n_k})\) that converges to a number in the set \(\mathbb R\) of reals.

\[\exists\,(n_k):\ \lim_{k\to\infty}a_{n_k}\in\mathbb R\]

One proof uses the monotone convergence theorem: every real sequence has a monotone subsequence (the peaks lemma), and if the sequence is bounded, that monotone subsequence is bounded and converges. Another proof is by bisection: split the interval \([-M,M]\) containing the sequence into two halves, pick one with infinitely many terms, and repeat; the nested intervals have length \(2M/2^j\to0\) and share a single point, which turns out to be the limit of a subsequence.

Boundedness cannot be dropped: \(a_n=n\) has no convergent subsequence. Nor can we expect the whole sequence to converge: \((-1)^n\) is bounded and diverges, but its even subsequence converges to \(1\) and its odd one to \(-1\). The theorem only asserts the existence of at least one convergent subsequence, not uniqueness.

A useful consequence: if a bounded sequence has a single value as the limit of convergent subsequences, the whole sequence converges to it. The theorem also depends essentially on completeness: it fails in \(\mathbb Q\), because the decimal truncations of \(\sqrt2\) are bounded and their subsequences converge to an irrational.

The theorem drives later results: the extreme value theorem, the equivalence between closed-and-bounded and compact in \(\mathbb R\), and the Cauchy criterion.

Think about it

What does the theorem say about \(a_n=\sin n\)?

Since \(|\sin n|\le1\), some subsequence converges, even if we cannot describe it explicitly.

A bounded sequence has two subsequences with different limits and diverges. Does this contradict the theorem?

No: the theorem only ensures that some subsequence converges, not that the whole sequence does.

Exercises 10

  1. Which of these sequences satisfies the hypothesis of the Bolzano-Weierstrass theorem?
  2. Let \(a_n=(-1)^n\). Which of these is a convergent subsequence?
  3. For \(a_n=n\), why does the Bolzano-Weierstrass theorem not guarantee a convergent subsequence?
  4. Let \(a_n=\frac{3n+1}{n+1}\). To what number does every subsequence of \((a_n)\) converge?
  5. Which of these statements is true?
  6. Let \(a_n=\sin\left(\frac{n\pi}{3}\right)\). How many distinct numbers are the limit of some subsequence?
  7. In the bisection proof we start with \([-8,8]\) and at each step keep the half with infinitely many terms. How long is the interval after 3 steps?
  8. If \((a_n)\) is bounded and all its convergent subsequences have limit \(4\), what follows?
  9. Why does the Bolzano-Weierstrass theorem fail if only rational numbers are allowed?
  10. Let \(a_n=(-1)^n\frac{n}{n+1}\). What is the largest limit of a convergent subsequence?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §2.5, Teorema 2.5.5
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 3, Teorema 3.6(b)
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §3.4, Teorema 3.4.7