Sequences · Definition

Cauchy sequence

real analysissequences

Before you start you need to know

It lets us decide whether a sequence converges without knowing the limit: we only look at how the terms behave relative to each other.

Definition

A sequence \((a_n)\) is Cauchy if, for every \(\varepsilon>0\), there is an index \(N\) beyond which any two terms are less than \(\varepsilon\) apart.

\[\forall\varepsilon>0\ \exists N:\ m,n\ge N\Rightarrow|a_m-a_n|<\varepsilon\]

Every convergent sequence is Cauchy: if \(a_n\to L\), for large \(m,n\) both terms are within \(\varepsilon/2\) of \(L\), and the triangle inequality gives \(|a_m-a_n|<\varepsilon\). The valuable part is the converse in \(\mathbb R\): every Cauchy sequence converges. Thus convergence and the Cauchy condition are equivalent, and the latter never mentions the limit.

The proof of the converse uses earlier results. A Cauchy sequence is bounded (beyond \(N\) all terms lie within \(1\) of \(a_N\)); by Bolzano–Weierstrass it has a subsequence converging to some \(L\); and the Cauchy property forces the whole sequence to converge to that \(L\).

Beware of a weaker condition: \(a_{n+1}-a_n\to0\) is not enough. The harmonic partial sums \(H_n\) satisfy \(H_{n+1}-H_n=\frac1{n+1}\to0\), yet \(H_{2n}-H_n\ge\frac12\), so they are not Cauchy and diverge. The definition requires the distance to be small for all pairs \(m,n\ge N\), not just neighbours.

The equivalence is a feature of the completeness of \(\mathbb R\). In \(\mathbb Q\) the decimal truncations of \(\sqrt2\) are Cauchy but have no rational limit. That is why the Cauchy notion is used to define complete spaces and, in later courses, to construct \(\mathbb R\) from \(\mathbb Q\).

Think about it

Show that \(a_n=\sqrt{n+1}-\sqrt n\) tends to \(0\) but \(s_n=\sum_{k=1}^n a_k\) is not Cauchy.

We have \(a_n=\frac1{\sqrt{n+1}+\sqrt n}\to0\). But the telescoping sum gives \(s_n=\sqrt{n+1}-1\), which is unbounded; hence it does not converge and is not Cauchy.

If \((a_n)\) and \((b_n)\) are Cauchy, is \((a_n+b_n)\)?

Yes: \(|(a_m+b_m)-(a_n+b_n)|\le|a_m-a_n|+|b_m-b_n|\), and each summand is less than \(\varepsilon/2\) for large \(m,n\).

Exercises 10

  1. Which of these sequences is Cauchy?
  2. For \(a_n=1/n\), what is the smallest \(N\) such that \(|a_m-a_n|<0.01\) whenever \(m,n\ge N\)?
  3. If \(a_n=1/n\), compute \(|a_{10}-a_{20}|\).
  4. Let \(a_n=1+\dfrac{(-1)^n}{n}\). Compute \(|a_3-a_4|\).
  5. Which statement is true in \(\mathbb R\)?
  6. Let \(H_n=1+\frac12+\cdots+\frac1n\). What is the smallest value of \(|H_{2n}-H_n|\) over \(n\ge1\)?
  7. If \(|a_{k+1}-a_k|\le 1/2^k\) for all \(k\), then \(|a_m-a_n|<1/2^{n-1}\) for \(m>n\). What is the smallest \(N\) this bound guarantees for \(|a_m-a_n|<\frac1{100}\) with \(m,n\ge N\)?
  8. Why is \(a_n=(-1)^n\) not Cauchy?
  9. A Cauchy sequence \((a_n)\) has a subsequence converging to \(L\). What follows?
  10. Let \(a_n\) be \(\sqrt2\) truncated to \(n\) decimals: \(1,\,1.4,\,1.41,\,1.414,\dots\). Viewed in \(\mathbb Q\), what happens?

References

  • Stephen Abbott, Understanding Analysis, 2nd edition, Springer, 2015. §2.6, Definición 2.6.1
  • Walter Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976. Cap. 3, Definición 3.12
  • Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, 4th edition, Wiley, 2011. §3.5, Definición 3.5.2