uniformCauchySeqOnFilter — Mathlib

English

If hf is a UniformCauchySeqOnFilter and hp: p'' ≤ p, then UniformCauchySeqOnFilter F p'' p' holds.

Русский

Если hf — UniformCauchySeqOnFilter и hp: p'' ≤ p, то UniformCauchySeqOnFilter F p'' p' выполняется.

LaTeX

$$$hf \to hp \,:\, UniformCauchySeqOnFilter F p'' p'$$$

Lean4

/-- A sequence that converges uniformly is also uniformly Cauchy -/
theorem uniformCauchySeqOnFilter (hF : TendstoUniformlyOnFilter F f p p') : UniformCauchySeqOnFilter F p p' :=
  by
  intro u hu
  rcases comp_symm_of_uniformity hu with ⟨t, ht, htsymm, htmem⟩
  have := tendsto_swap4_prod.eventually ((hF t ht).prod_mk (hF t ht))
  apply this.diag_of_prod_right.mono
  simp only [and_imp, Prod.forall]
  intro n1 n2 x hl hr
  exact Set.mem_of_mem_of_subset (prodMk_mem_compRel (htsymm hl) hr) htmem

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