English
If oscillation f x is less than ε for all x in a compact K, then a global bound on the oscillation over balls around K holds with some δ.
Русский
Если untuk все x в K компактном f(x) меньше ε, тогда существует δ>0, на шаре вокруг K осцилляция ограничена.
LaTeX
$$$\\text{IsCompact } K \\Rightarrow (∀ x∈K, \\operatorname{oscillationWithin} f x < ε) → ∃ δ>0, ∀ x∈K, \\operatorname{diam}(f(B(x,δ))) ≤ ε$$$
Lean4
theorem sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n) (hu : Monotone u) (n : ℕ) :
(∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k := by
induction n with
| zero => simp
| succ n
ihn =>
suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k
by
rw [sum_range_succ, ← sum_Ico_consecutive]
exacts [add_le_add ihn this, (add_le_add_right (hu n.zero_le) _ : u 0 + 1 ≤ u n + 1),
add_le_add_right (hu n.le_succ) _]
have : ∀ k ∈ Ico (u n + 1) (u (n + 1) + 1), f (u (n + 1)) ≤ f k := fun k hk =>
hf (Nat.lt_of_le_of_lt (Nat.succ_le_of_lt (h_pos n)) <| (Nat.lt_succ_of_le le_rfl).trans_le (mem_Ico.mp hk).1)
(Nat.le_of_lt_succ <| (mem_Ico.mp hk).2)
convert sum_le_sum this
simp
