English
In a normed field, CauSeq is equivalent to usual Cauchy convergence; the Tendsto statement holds in the filter sense.
Русский
В нормированном поле CauSeq эквивалентно обычной схождению, утверждение о пределе выполняется в виде предела по фильтру.
LaTeX
$$$ \mathrm{tendsto\,limit}(f) \text{ in Filter atTop} $$$
Lean4
theorem dist_lt (hA : VanishingDiam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β) :
∃ n : ℕ, ∀ (y) (_ : y ∈ A (res x n)) (z) (_ : z ∈ A (res x n)), dist y z < ε :=
by
specialize hA x
rw [ENNReal.tendsto_atTop_zero] at hA
obtain ⟨n, hn⟩ := hA (ENNReal.ofReal (ε / 2)) (by simp only [gt_iff_lt, ENNReal.ofReal_pos]; linarith)
use n
intro y hy z hz
rw [← ENNReal.ofReal_lt_ofReal_iff ε_pos, ← edist_dist]
apply lt_of_le_of_lt (EMetric.edist_le_diam_of_mem hy hz)
apply lt_of_le_of_lt (hn _ (le_refl _))
rw [ENNReal.ofReal_lt_ofReal_iff ε_pos]
linarith
