Theorem seq1hcau

Description: A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999) (Revised by Mario Carneiro, 14-May-2014) (New usage is discouraged.)

	Ref	Expression
Assertion	seq1hcau	⊢ ( 𝐹 : ℕ ⟶ ℋ → ( 𝐹 ∈ Cauchy ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		hcau	⊢ ( 𝐹 ∈ Cauchy ↔ ( 𝐹 : ℕ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) )
2	1	baib	⊢ ( 𝐹 : ℕ ⟶ ℋ → ( 𝐹 ∈ Cauchy ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) )