Cauchy Sequences, Completeness, and Convergence Tests

All flashcards

Cauchy sequence in a metric space

A sequence $(x_n)$ such that for every $\varepsilon>0$, some $N$ satisfies $d(x_m,x_n)<\varepsilon$ whenever $m,n>N$.

Modulus of Cauchy convergence

A function giving, for each requested precision, an index after which all pairs of terms meet that precision.

Complete metric space

A metric space in which every Cauchy sequence converges to a point in that space.

Uniform continuity and Cauchy sequences

A uniformly continuous function maps Cauchy sequences in its domain to Cauchy sequences in its codomain.

Completion of a metric space

A complete space containing the original space as a dense subspace and supplying limits for its Cauchy sequences.

Quiz questions

1. For $a_n=\sqrt{n}$, the consecutive differences tend to zero. Why is $(a_n)$ nevertheless not Cauchy?Answer: The sequence is unbounded, whereas every Cauchy sequence in a metric space is bounded.
2. Why is every convergent sequence in a metric space Cauchy?Answer: Late terms are each within $\varepsilon/2$ of the limit, so the triangle inequality puts them less than $\varepsilon$ apart.
3. Which sequence illustrates that $\mathbb{Q}$ is incomplete under the usual distance?Answer: Rational decimal truncations of $\sqrt{2}$ form a Cauchy sequence with no limit in $\mathbb{Q}$.
4. For a real series with terms $a_n$, what condition on its tails is equivalent to convergence?Answer: For every $\varepsilon>0$, sums $\sum_{j=q+1}^{p}a_j$ become smaller than $\varepsilon$ in absolute value whenever $p>q$ are sufficiently large.
5. In a set with the discrete metric, what must a Cauchy sequence eventually do?Answer: It must be constant from some index onward.