English
If f is monotone on s, a is IsLUB s a, s nonempty, and f tends to b along nhdsWithin a s, then IsLUB (f'' s) b.
Русский
Если f монотонна на s, a является IsLUB s a, s непусто, и f сходится к b по nhdsWithin a s, тогда IsLUB (f'' s) b.
LaTeX
$$MonotoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (nhdsWithin a s) (nhds b) → IsLUB (f '' s) b$$
Lean4
theorem isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α}
{b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) :
IsLUB (f '' s) b :=
haveI := ha.nhdsWithin_neBot hs
⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' =>
le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <| mem_image_of_mem _ hx)⟩
