English
If a sequence tends to x, the vonNBoundedness of its image is preserved under range.
Русский
Если последовательность сходится к x, ограниченность образа сохраняется под диапазоном.
LaTeX
$$$\\operatorname{Tendsto}(f,\\; atTop,\\; \\mathcal{N}(x)) \\Rightarrow \\operatorname{IsVonNBounded}_{\\mathbb{k}}(\\operatorname{range}(f))$$$
Lean4
theorem isBounded_iff_subset_smul_ball {s : Set E} : Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.ball (0 : E) 1 :=
by
rw [← isVonNBounded_iff 𝕜]
constructor
· intro h
rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with ⟨ρ, _, hρball⟩
rcases NormedField.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩
exact ⟨a, hρball a ha.le⟩
· rintro ⟨a, ha⟩
exact ((isVonNBounded_ball 𝕜 E 1).image (a • (1 : E →L[𝕜] E))).subset ha