English
For nonzero exponent n, the infimum of a powers preserves exponentiation: (sup f)^n equals sup of f^n.
Русский
Если показатель n \u2260 0, инфимум (sup f)^n равно над инфимумом f^n.
LaTeX
$$⨅ i, f(i)^n = (⨅ i, f(i))^n for n ≠ 0$$
Lean4
/-- An antitone sequence `f : ℕ → ℝ≥0` has a finite limit. -/
theorem tendsto_of_antitone {f : ℕ → ℝ≥0} (h_ant : Antitone f) : ∃ r : ℝ≥0, Tendsto f atTop (𝓝 r) :=
by
have h_bdd_0 : (0 : ℝ) ∈ lowerBounds (Set.range fun n : ℕ => (f n : ℝ)) :=
by
rintro r ⟨n, hn⟩
simp_rw [← hn]
exact NNReal.coe_nonneg _
obtain ⟨L, hL⟩ := Real.tendsto_of_bddBelow_antitone ⟨0, h_bdd_0⟩ h_ant
have hL0 : 0 ≤ L :=
haveI h_glb : IsGLB (Set.range fun n => (f n : ℝ)) L := isGLB_of_tendsto_atTop h_ant hL
(le_isGLB_iff h_glb).mpr h_bdd_0
exact ⟨⟨L, hL0⟩, NNReal.tendsto_coe.mp hL⟩
