Why is testing only consecutive terms insufficient for the Cauchy condition?

Testing only consecutive terms is insufficient because the Cauchy condition requires that *every* pair of sufficiently late terms is close, not just adjacent ones. A sequence can have consecutive differences approaching zero while still diverging or failing to cluster tightly over long intervals. The classic counterexample is the sequence

a_n = \log n

, where

|a_{n+1} - a_n| \to 0

n \to \infty

, but the sequence itself diverges to infinity and is not Cauchy.

Conditions

The sequence is defined over the real numbers.
The condition checks $|a_{n+1} - a_n| \to 0$ .
The Cauchy condition requires $\forall \varepsilon > 0, \exists N, \forall m, n > N: |a_m - a_n| < \varepsilon$ .

Reasoning, step by step

Understand that the Cauchy condition quantifies over all pairs $m, n > N$ .
Recognize that consecutive terms are a specific subset of these pairs ( $m = n+1$ ).
Identify that controlling only the subset does not control the entire set of pairs.
Use the counterexample $a_n = \log n$ to show that $|a_{n+1} - a_n| \to 0$ does not imply the sequence is Cauchy.
Conclude that the full pairwise condition is necessary to ensure the tail of the sequence is uniformly bounded in diameter.

Example

The card 'Consecutive differences are insufficient' states: 'For example log n has consecutive differences tending to zero but diverges. The Cauchy condition controls all pairs in a tail.'

Common misconceptions

Assuming that if $|a_{n+1} - a_n| < \varepsilon$ for large $n$ , then $|a_m - a_n| < \varepsilon$ for all $m, n > N$ .
Believing that small steps imply the path cannot wander far from its starting point in the tail.

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Why is testing only consecutive terms insufficient for the Cauchy condition?

Conditions

Reasoning, step by step

Example

Common misconceptions

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Connected concepts

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Related questions

Why is testing only consecutive terms insufficient for the Cauchy condition?

Conditions

Reasoning, step by step

Example

Common misconceptions

Watch the explanation

Connected concepts

Explore next

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