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$x_k = \sqrt{k}$ has $x_{k+1} - x_k \to 0$ and is not Cauchy

Statement refuted

Refuted claim: a sequence of reals whose consecutive differences tend to $0$ is Cauchy (FALSE: if $|x_{k+1} - x_k| \to 0$ then $(x_k)$ is Cauchy, Limits and Cauchy sequences of reals).

The witness is $x_k = \sqrt k$ for $k \in \mathbb{N}$ . Its consecutive differences satisfy

$x_{k+1} - x_k \;=\; \sqrt{k+1} - \sqrt{k} \;=\; \frac{1}{\sqrt{k+1} + \sqrt{k}} \;\longrightarrow\; 0,$

while the sequence itself is unbounded, hence not Cauchy (Every Cauchy sequence of reals is bounded). The refutation is carried out in full in FALSE: if $|x_{k+1} - x_k| \to 0$ then $(x_k)$ is Cauchy; this item records the witness and adds the sharper statement that $\sqrt k$ diverges to $+\infty$ (Divergence to $+\infty$ and to $-\infty$ ).

Facts & Assumptions

Given: The sequence $(x_k)$ of reals with $x_k := \sqrt k$ , where $k$ denotes the canonical natural $k \cdot 1_{\mathbb{R}}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences, Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$ ; the positives are $\{x^2 : x \neq 0\}$ ).

[L1]

The witness and its two properties: $x_{k+1} - x_k = 1/(\sqrt{k+1} + \sqrt k)$ tends to $0$ , and $(x_k)$ is unbounded and not Cauchy (FALSE: if $|x_{k+1} - x_k| \to 0$ then $(x_k)$ is Cauchy).

[L2]

Square roots, and the factorisation $b^2 - a^2 = (b-a)(b+a)$ (Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$ ; the positives are $\{x^2 : x \neq 0\}$ , Factorisation of $b^n - a^n$ , and the resulting Lipschitz estimate, Integer powers $a^m$ ).

[L3]

Powers and order: for $a, b \ge 0$ and $n \ge 1$ , $a < b$ exactly when $a^n < b^n$ (Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$ ).

[L4]

Canonical naturals: positive for $n \ge 1$ , and strictly increasing in the index (Canonical naturals are positive and strictly increasing); reciprocals of positives are positive and reciprocation reverses the order (Inverses of positives are positive, and reciprocation reverses order).

[L5]

Archimedean property, both forms (Every complete ordered field is Archimedean, For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ).

[L6]

Absolute value: $|t| = t$ for $t \ge 0$ , and $|t| \ge t$ (Basic properties of the absolute value).

[L7]

Every Cauchy sequence of reals is bounded (Every Cauchy sequence of reals is bounded).

[L8]

Divergence to $+\infty$ : for every real $M$ there is $K$ with $x_k > M$ for all $k \ge K$ (Divergence to $+\infty$ and to $-\infty$ ).

[L9]

Trichotomy and transitivity of the order on $\mathbb{R}$ (Complete ordered field (least-upper-bound property), Ordered field).

Counterexample

technique · direct

1.1

The sequence $(x_k)$ satisfies the hypothesis of the refuted claim, its consecutive differences tending to $0$ , and it is not Cauchy.

givenL1L7

1.2

The failure is as strong as possible: $(x_k)$ diverges to $+\infty$ . Let $M \in \mathbb{R}$ and put $M' := |M| \ge M$ , so $M' \ge 0$ . By [L5] fix a natural $n \ge 1$ with $(M')^2 < n$ .

givenL5L6

2.1

It therefore refutes the claim: having null consecutive differences does not make a sequence Cauchy.

step 1.1L1

3.1

For every $k \ge n$ : $(x_k)^2 = k \ge n > (M')^2$ with $x_k \ge 0$ and $M' \ge 0$ , so $x_k > M' \ge M$ . Since $M$ was arbitrary, $x_k \to +\infty$ .

step 1.2L2L3L4L8L9∎

Remarks

The gaps are null and their sums are not. The differences are about $1/(2\sqrt k)$ , so they tend to $0$ ; but they telescope, and $x_m - x_k = \sqrt m - \sqrt k$ is as large as one likes for $m$ large. The Cauchy condition constrains $|x_m - x_k|$ for all large pairs, and no hypothesis about consecutive pairs alone can deliver that.
What the correct hypothesis looks like. Geometric decay of the gaps, with a single ratio $c < 1$ at every index, does suffice (Contractive sequence: $|x_{k+2} - x_{k+1}| \le c\,|x_{k+1} - x_k|$ for a fixed $0 < c < 1$ , Every contractive sequence is Cauchy, hence converges, with error bound $|x - x_k| \le c^{k-1}|x_2 - x_1|/(1-c)$ for $k \ge 1$ ), because then the telescoped sums are dominated by a convergent geometric bound. Merely shrinking gaps are not enough either, which is $x_{k+1} = x_k + 1/x_k$ from $x_1 = 1$ has strictly decreasing consecutive gaps and diverges, so no uniform $c < 1$ exists.
The same sequence separates two notions that are easy to confuse. Its gaps are null, so it is "eventually almost constant" in a naive reading; it is nevertheless unbounded and divergent to $+\infty$ . Nothing about the local behaviour of a sequence controls its global behaviour.

Depends on

Used by

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Sources

Cauchy sequence (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)
J. Lebl, Basic Analysis I, §2.4 (standard reference, not scraped)
Sequence of Square Roots of Natural Numbers is not Cauchy (ProofWiki) (standard reference, not scraped)

xk=kx_k = \sqrt{k}xk​=k​ has xk+1−xk→0x_{k+1} - x_k \to 0xk+1​−xk​→0 and is not Cauchy