False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-26

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FALSE: $\limsup(x_k + y_k) = \limsup x_k + \limsup y_k$

Statement

False claim: for all sequences $(x_k)$ , $(y_k)$ of reals (Sequences of reals: bounded, eventually, frequently, tails, subsequences) whose limit superiors have a defined sum in $\overline{\mathbb{R}}$ (The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$ , its order, and the arithmetic that is left undefined),

$\limsup_{k}(x_k + y_k) \;=\; \limsup_{k} x_k + \limsup_{k} y_k .$

The corresponding statement with $=$ replaced by $\le$ is true and is $\limsup(x_k + y_k) \le \limsup x_k + \limsup y_k$ whenever the right-hand side is defined in $\overline{\mathbb{R}}$ , and dually for $\liminf$ . The claim above is what one gets by strengthening that inequality to an equality, and it fails: the two sides can differ by as much as the whole oscillation of the sequences, because the two limit superiors may be attained along different sets of indices while the sum of the sequences never sees either of them.

The witness is $x_k = (-1)^k$ and $y_k = -(-1)^k$ , refuted below; it is recorded separately as a named counterexample on the companion page.

Facts & Assumptions

Given: The alternating sequence $(s_k)$ and the index maps $e, o$ of The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $\mathbb{N}$ their disjoint union, and the unique $(s_k)$ with $s_0 = 1$ , $s_{\sigma(k)} = -s_k$ , which satisfies $|s_k| = 1$ , $s \circ e \equiv 1$ and $s \circ o \equiv -1$ ; the sequences $x_k := s_k$ and $y_k := -s_k$ ; and the tail ranges and extended tail bounds of Limit superior and limit inferior of a real sequence as $\inf_n \sup_{k \ge n} x_k$ and $\sup_n \inf_{k \ge n} x_k$ in $\overline{\mathbb{R}}$ .

[L1]

The alternating sequence: $|s_k| = 1$ for every $k$ , $s_{e_j} = 1$ and $s_{o_j} = -1$ for every $j$ , and $e$ , $o$ are strictly increasing (The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $\mathbb{N}$ their disjoint union, and the unique $(s_k)$ with $s_0 = 1$ , $s_{\sigma(k)} = -s_k$ , which satisfies $|s_k| = 1$ , $s \circ e \equiv 1$ and $s \circ o \equiv -1$ ).

[L2]

A strictly increasing index map satisfies $n_j \ge j$ (A strictly increasing index map satisfies $n_k \ge k$ ).

[L3]

Limit superior: $\limsup_k z_k = \inf\{\sup T_n(z)\}$ with $T_n(z) = \{z_k : k \ge n\}$ ; all these bounds exist in $\overline{\mathbb{R}}$ , $\sup$ being the least upper bound and $\inf$ the greatest lower bound (Limit superior and limit inferior of a real sequence as $\inf_n \sup_{k \ge n} x_k$ and $\sup_n \inf_{k \ge n} x_k$ in $\overline{\mathbb{R}}$ , The tail suprema of any real sequence are nonincreasing in $\overline{\mathbb{R}}$ , so the limit superior exists for every sequence, Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$ , Upper bound, least upper bound, and strict upper bound, Partial order and partially ordered set).

[L4]

The order on $\overline{\mathbb{R}}$ is total and restricts on $\mathbb{R}$ to the order of $\mathbb{R}$ ; $\pm\infty$ are not real (The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$ , its order, and the arithmetic that is left undefined).

[L5]

Absolute value: $|t| = 1$ forces $t = 1$ or $t = -1$ (Basic properties of the absolute value, Absolute value in an ordered field).

[L6]

Order arithmetic: $0 < 1$ , so $-1 < 1$ and $0 < 1 + 1 = 2$ ; in particular $0 \ne 2$ (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)).

[L7]

Subadditivity: $\limsup_k(z_k + w_k) \le \limsup_k z_k + \limsup_k w_k$ whenever the right-hand side is defined ( $\limsup(x_k + y_k) \le \limsup x_k + \limsup y_k$ whenever the right-hand side is defined in $\overline{\mathbb{R}}$ , and dually for $\liminf$ ).

[L8]

The refuted claim: for all sequences whose limit superiors have a defined sum, $\limsup_k(z_k + w_k) = \limsup_k z_k + \limsup_k w_k$ .

Refutation

technique · direct

1.1

The sequences $x_k = s_k$ and $y_k = -s_k$ are sequences of reals, and $x_k + y_k = s_k + (-s_k) = 0$ for every $k$ .

givenL1

1.2

Every value $s_k$ is $1$ or $-1$ , since $|s_k| = 1$ ; and for every $n$ both values occur at an index $\ge n$ , since $s_{e_n} = 1$ with $e_n \ge n$ and $s_{o_n} = -1$ with $o_n \ge n$ .

givenL1L2L5

2.1

Hence $T_n(x) = \{1, -1\}$ for every $n$ . Its least upper bound in $\overline{\mathbb{R}}$ is $1$ : the element $1$ bounds both $1$ and $-1$ from above because $-1 < 1$ , and any upper bound $u$ satisfies $1 \le u$ because $1 \in T_n(x)$ . So $\sup T_n(x) = 1$ for every $n$ , and $\limsup_k x_k$ is the greatest lower bound of the one-element family $\{1\}$ , namely $1$ .

step 1.2L3L4L6

2.2

The sequence $y_k = -s_k$ takes the value $1$ at every $o_n$ and the value $-1$ at every $e_n$ , and takes no other value, so $T_n(y) = \{1, -1\}$ for every $n$ as well, and the same computation gives $\limsup_k y_k = 1$ .

step 1.2L1L2L3L4L5L6

2.3

The sum sequence is constantly $0$ , so $T_n(x+y) = \{0\}$ , whose least upper bound is $0$ , and $\limsup_k(x_k + y_k) = 0$ .

step 1.1L3L4

3.1

Both limit superiors are the real number $1$ , so their sum is defined and equals $1 + 1 = 2$ , and the claim asserts $0 = 2$ for this pair. But $0 < 2$ , so $0 \ne 2$ and the claim fails.

step 2.1step 2.2step 2.3L6L8

4.1

The claim is therefore false. What survives is the inequality of [L7], which for this pair reads $0 \le 2$ and is strict.

step 3.1L7L8∎

Remarks

Which half of the equality fails. Only $\ge$ ; the inequality $\le$ is a theorem ( $\limsup(x_k + y_k) \le \limsup x_k + \limsup y_k$ whenever the right-hand side is defined in $\overline{\mathbb{R}}$ , and dually for $\liminf$ ). So the claim is not wrong by accident of the witness: the reverse inequality has no proof, and this pair shows it has no proof because it is false.
The mechanism is a mismatch of index sets. $\limsup_k x_k = 1$ is achieved along the even indices and $\limsup_k y_k = 1$ along the odd ones. The sum can only see both at once if some index is large in both sequences simultaneously, and here no index is: where $x_k = 1$ one has $y_k = -1$ . Equality does hold when one of the two sequences converges, because then its limit superior is achieved along every subsequence.
The witness is named on the companion page as $x_k = (-1)^k$ , $y_k = (-1)^{k+1}$ give $\limsup(x_k + y_k) = 0 < 2 = \limsup x_k + \limsup y_k$ ↗, which quotes the computation made here.
The defined-sum hypothesis is inherited from $\limsup(x_k + y_k) \le \limsup x_k + \limsup y_k$ whenever the right-hand side is defined in $\overline{\mathbb{R}}$ , and dually for $\liminf$ and is not what fails here. Both limit superiors in the witness are real, so the right-hand side is a perfectly good real number; the equality is false anyway.

Depends on

Used by

xₖ = (-1)ᵏ, yₖ = (-1)ᵏ⁺¹ give limsup(xₖ + yₖ) = 0 < 2 = limsup xₖ + limsup yₖ Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 72 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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 (standard reference, not scraped)

FALSE: lim sup⁡(xk+yk)=lim sup⁡xk+lim sup⁡yk\limsup(x_k + y_k) = \limsup x_k + \limsup y_klimsup(xk​+yk​)=limsupxk​+limsupyk​