TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-26

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A real sequence converges to $L \in R$ iff $lim inf x_{k} = lim sup x_{k} = L$ , and diverges to $\pm \infty$ iff both equal $\pm \infty$

Statement

Let $(x_{k})$ be a sequence of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences), with $lim sup$ and $lim inf$ as in Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ .

For $L \in R$ : $(x_{k})$ converges to $L$ (Limits and Cauchy sequences of reals) if and only if ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = L$ .
$x_{k} \to + \infty$ (Divergence to $+ \infty$ and to $- \infty$ ) if and only if ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = + \infty$ . Moreover ${lim inf}_{k} x_{k} = + \infty$ on its own already forces ${lim sup}_{k} x_{k} = + \infty$ .
$x_{k} \to - \infty$ if and only if ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = - \infty$ , and ${lim sup}_{k} x_{k} = - \infty$ on its own already forces ${lim inf}_{k} x_{k} = - \infty$ .

The three clauses combine into one statement about the extended line: for $L \in \overline{R}$ , the sequence $(x_{k})$ converges to $L$ in $\overline{R}$ (Convergence in $\overline{R}$ and the extended subsequential limit set: $L \in \overline{R}$ is an extended subsequential limit when some subsequence converges to $L$ , or diverges to $L = \pm \infty$ ) if and only if

${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = L .$

Since ${lim inf}_{k} x_{k} \leq {lim sup}_{k} x_{k}$ always ( $lim inf x_{k} \leq lim sup x_{k}$ for every real sequence), the single equation ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k}$ is therefore equivalent to convergence in $\overline{R}$ , and the common value is the limit. A sequence that neither converges nor diverges to $\pm \infty$ is exactly one for which the inequality is strict.

Facts & Assumptions

Given: A sequence $(x_{k})$ of reals, its tail ranges $T_{n} = {x_{k} : k \geq n}$ , the extended tail bounds $s_{n} = \sup T_{n}$ and $i_{n} = \inf T_{n}$ , and the quantities ${lim sup}_{k} x_{k} = \inf {s_{n}}$ , ${lim inf}_{k} x_{k} = \sup {i_{n}}$ (Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ ).

[L1]

All of $s_{n}$ , $i_{n}$ , ${lim sup}_{k} x_{k}$ and ${lim inf}_{k} x_{k}$ exist in $\overline{R}$ for every sequence; $i_{n}$ is the greatest lower bound of $T_{n}$ and ${lim inf}_{k} x_{k}$ the least upper bound of ${i_{n}}$ , with the dual descriptions for $s_{n}$ and ${lim sup}_{k} x_{k}$ (The tail suprema of any real sequence are nonincreasing in $\overline{R}$ , so the limit superior exists for every sequence, Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ , Upper bound, least upper bound, and strict upper bound, Partial order and partially ordered set).

[L2]

The order on $\overline{R}$ is total, so the failure of $a \leq b$ is $b < a$ ; it restricts on $R$ to the order of $R$ ; $+ \infty$ is the greatest element and $- \infty$ the least; and every real is $< + \infty$ and $> - \infty$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined, Partial order and partially ordered set).

[L3]

Epsilon characterisation, for a real $L$ : $L = {lim sup}_{k} x_{k}$ exactly when for every real $ε > 0$ one has $x_{k} < L + ε$ eventually and $x_{k} > L - ε$ frequently; and $L = {lim inf}_{k} x_{k}$ exactly when for every real $ε > 0$ one has $x_{k} > L - ε$ eventually and $x_{k} < L + ε$ frequently (For finite $L$ : $L = lim sup x_{k}$ iff for every $ε > 0$ one has $x_{k} < L + ε$ eventually and $x_{k} > L - ε$ frequently).

[L4]

${lim inf}_{k} x_{k} \leq {lim sup}_{k} x_{k}$ ( $lim inf x_{k} \leq lim sup x_{k}$ for every real sequence).

[L5]

Reflection: ${lim sup}_{k} (- x_{k}) = - {lim inf}_{k} x_{k}$ and ${lim inf}_{k} (- x_{k}) = - {lim sup}_{k} x_{k}$ ( $lim sup (- x_{k}) = - lim inf (x_{k})$ , with the reflection of $\overline{R}$ exchanging $\pm \infty$ ). Also $x_{k} \to - \infty$ if and only if $- x_{k} \to + \infty$ : the condition $x_{k} < M$ for all $k \geq K$ is equivalent to $- x_{k} > - M$ for all $k \geq K$ by order reversal, and $M$ runs over all reals exactly when $- M$ does (Divergence to $+ \infty$ and to $- \infty$ ); the order reversal used here is strict, and the form stated in Order is preserved by adding a constant and by adding inequalities is likewise strict, so nothing nonstrict is being borrowed from it.

[L6]

Convergence to a real $L$ means: for every rational $ε > 0$ there is $K$ with $∣ x_{k} - L ∣ < ε$ for all $k \geq K$ ; and the same relation is obtained by testing every real $ε > 0$ instead, since below any positive real lies a positive rational (Limits and Cauchy sequences of reals, Sequences of reals: bounded, eventually, frequently, tails, subsequences, The rationals embed densely in the reals).

[L7]

Divergence: $x_{k} \to + \infty$ means that for every real $M$ there is $K$ with $x_{k} > M$ for all $k \geq K$ (Divergence to $+ \infty$ and to $- \infty$ ).

[L8]

Eventually and frequently, and the fact that a property holding eventually holds frequently, since indices beyond any two given thresholds exist by totality of the order on $N$ ; likewise two properties each holding eventually hold together from the larger threshold on (Sequences of reals: bounded, eventually, frequently, tails, subsequences, $\leq$ is a linear order on $N$ ).

[L9]

Absolute value: for $c > 0$ , $∣ a ∣ < c$ if and only if $- c < a < c$ (Basic properties of the absolute value).

[L10]

Order arithmetic in $R$ : $0 < 1$ , so $t < t + 1$ for every real $t$ , and no real is above every real; adding a constant preserves the order (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Ordered field, Complete ordered field (least-upper-bound property)).

Proof

technique · direct

1.1

For the forward implication of claim 1, assume $L \in R$ and that $(x_{k})$ converges to $L$ .

assume-hypL6

1.2

For the converse implication of claim 1, assume $L \in R$ and ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = L$ .

assume-hypL1

1.3

For the forward implication of claim 2, assume $x_{k} \to + \infty$ .

assume-hypL7

1.4

For the converse implication of claim 2, assume ${lim inf}_{k} x_{k} = + \infty$ .

assume-hypL1

2.1

Under the assumption of step 1.1, let $ε > 0$ be an arbitrary real. Testing convergence at $ε$ gives $K$ with $∣ x_{k} - L ∣ < ε$ , hence $L - ε < x_{k} < L + ε$ , for all $k \geq K$ . So $x_{k} < L + ε$ eventually and $x_{k} > L - ε$ eventually, and each of the two therefore also holds frequently. Both halves of each characterisation in [L3] are met, so ${lim sup}_{k} x_{k} = L$ and ${lim inf}_{k} x_{k} = L$ .

step 1.1L3L6L8L9

2.2

Under the assumption of step 1.2, let $ε > 0$ be an arbitrary real. The forward halves of the two characterisations in [L3] give $x_{k} < L + ε$ for all $k$ beyond some $K_{1}$ and $x_{k} > L - ε$ for all $k$ beyond some $K_{2}$ ; beyond the larger of $K_{1}$ and $K_{2}$ both hold, so $∣ x_{k} - L ∣ < ε$ there. This holds for every real $ε > 0$ , in particular for every rational one, so $(x_{k})$ converges to $L$ .

step 1.2L3L6L8L9

2.3

Under the assumption of step 1.3, let $M$ be an arbitrary real and take $K$ with $x_{k} > M$ for all $k \geq K$ . Then $M$ is a lower bound of $T_{K}$ , so $M \leq i_{K}$ , and $i_{K} \leq {lim inf}_{k} x_{k}$ because ${lim inf}_{k} x_{k}$ is an upper bound of ${i_{n}}$ ; hence $M \leq {lim inf}_{k} x_{k}$ . Since $M$ was an arbitrary real, ${lim inf}_{k} x_{k}$ is not $- \infty$ , which lies below every real, and it is not a real $t$ either, since $M = t + 1$ would give $t + 1 \leq t$ . So ${lim inf}_{k} x_{k} = + \infty$ .

step 1.3L1L2L7L10

2.4

Under the assumption of step 1.4, let $M$ be an arbitrary real. Since $\sup {i_{n}} = + \infty$ and $M < + \infty$ , the real $M$ is not an upper bound of ${i_{n}}$ , for otherwise the least upper bound would satisfy $+ \infty \leq M$ ; by totality there is $n$ with $i_{n} > M$ . Every $k \geq n$ satisfies $x_{k} \geq i_{n} > M$ , so $x_{k} > M$ eventually. As $M$ was arbitrary, $x_{k} \to + \infty$ .

step 1.4L1L2L7

3.1

Steps 2.1 and 2.2 are the two implications of claim 1.

step 2.1step 2.2L3

3.2

For claim 2: if $x_{k} \to + \infty$ then ${lim inf}_{k} x_{k} = + \infty$ by step 2.3, and then $+ \infty = {lim inf}_{k} x_{k} \leq {lim sup}_{k} x_{k}$ forces ${lim sup}_{k} x_{k} = + \infty$ since $+ \infty$ is the greatest element; conversely if ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = + \infty$ then in particular ${lim inf}_{k} x_{k} = + \infty$ and step 2.4 gives $x_{k} \to + \infty$ . The same use of [L4] is the additional assertion that ${lim inf}_{k} x_{k} = + \infty$ alone forces ${lim sup}_{k} x_{k} = + \infty$ .

step 2.3step 2.4L2L4

4.1

For claim 3, reflection gives $x_{k} \to - \infty$ exactly when $- x_{k} \to + \infty$ , which by claim 2 holds exactly when ${lim inf}_{k} (- x_{k}) = {lim sup}_{k} (- x_{k}) = + \infty$ , that is $- {lim sup}_{k} x_{k} = - {lim inf}_{k} x_{k} = + \infty$ , that is ${lim sup}_{k} x_{k} = {lim inf}_{k} x_{k} = - \infty$ ; and ${lim sup}_{k} x_{k} = - \infty$ alone forces ${lim inf}_{k} x_{k} \leq - \infty$ , hence ${lim inf}_{k} x_{k} = - \infty$ , since $- \infty$ is least. Claims 1, 2 and 3 together say that for $L \in \overline{R}$ the sequence converges to $L$ in $\overline{R}$ exactly when ${lim inf}_{k} x_{k} = {lim sup}_{k} x_{k} = L$ , since the three clauses of that definition are convergence to a real $L$ , divergence to $+ \infty$ and divergence to $- \infty$ .

step 3.1step 3.2L2L4L5∎

Remarks

This is the theorem that makes $lim sup$ and $lim inf$ worth defining. They exist for every sequence, with no hypothesis, and their coincidence is exactly convergence in $\overline{R}$ . So a question about convergence becomes a question about two computable quantities, and a proof of convergence can be assembled from one-sided estimates without a candidate limit in hand.
The equation is between elements of $\overline{R}$ , and reading it in $R$ would lose two thirds of the content. Clauses 2 and 3 are statements about divergence, and they are true statements about Divergence to $+ \infty$ and to $- \infty$ , not a redefinition of it: nothing above claims that a sequence diverging to $+ \infty$ has a limit in $R$ , and the symbol $+ \infty$ occurring in them is the element of $\overline{R}$ introduced in The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined.
A sequence with $lim inf < lim sup$ does neither. The alternating sequence is the standard witness, with the two values $- 1$ and $1$ ( $(- 1)^{k}$ has $lim inf = - 1$ and $lim sup = 1$ , so it does not converge ↗); it is bounded, so it also does not diverge to $\pm \infty$ , and the theorem says its failure to converge is exactly the gap between the two quantities.

Depends on

Used by

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51 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Limit superior and limit inferior (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (3.17) (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed., §6.4 (standard reference, not scraped)
J. Lebl, Basic Analysis I, §2.3 (standard reference, not scraped)

A real sequence converges to L∈RL∈R iff lim inf⁡xk=lim sup⁡xk=Lliminfxk​=limsupxk​=L, and diverges to ±∞±∞ iff both equal ±∞±∞