$x_k = 1 + (-1)^k$, $y_k = 1 + (-1)^{k+1}$ give $\limsup(x_k y_k) = 0 < 4$

Given: The alternating sequence $(s_k)$ 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 $s \circ o \equiv -1$ and the index maps $e, o$; the sequences $x_k := 1 + s_k$ and $y_k := 1 - s_k$, which are the families usually written $1 + (-1)^k$ and $1 + (-1)^{k+1}$; and the tail ranges $T_n$ 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 $\overline{\mathbb{R}}$.

The alternating sequence: $|s_k| = 1$ for every $k$, $s_{e_j} = 1$ and $s_{o_j} = -1$, and $e$, $o$ are strictly increasing with $n_j \ge j$ (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 $s \circ o \equiv -1$, A strictly increasing index map satisfies $n_k \ge k$).

Limit superior in $\overline{\mathbb{R}}$: existence for every sequence, and the least-upper-bound and greatest-lower-bound descriptions of the tail bounds (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, The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

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

Order and field arithmetic: $0 < 1$, so $0 < 2 = 1 + 1$ and $0 < 4 = 2 \cdot 2$; $1 + 1 = 2$, $1 - 1 = 0$, $1 + (-1) = 0$ and $1 - (-1) = 2$; a product with a zero factor is $0$ (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)).

Submultiplicativity: for bounded nonnegative sequences, $\limsup_k(x_k y_k) \le (\limsup_k x_k)(\limsup_k y_k)$, all three quantities being real (For bounded nonnegative sequences, $\limsup(x_k y_k) \le (\limsup x_k)(\limsup y_k)$).