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

1.1assume-hypL6

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

1.2assume-hypL1

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

1.3assume-hypL7

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

1.4assume-hypL1

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

2.1step 1.1L3L6L8L9

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

2.2step 1.2L3L6L8L9

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 $L$ .

2.3step 1.3L1L2L7L10

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 ${lim inf}_{k} x_{k} = + \infty$ .

2.4step 1.4L1L2L7

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, $x_{k} \to + \infty$ .

3.1step 2.1step 2.2L3

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

3.2step 2.3step 2.4L2L4

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 ${lim sup}_{k} x_{k} = + \infty$ .

4.1step 3.1step 3.2L2L4L5∎

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 $- \infty$ .

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.

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

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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 $\pm \infty$