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Metamath Proof Explorer

Theorem hcauseq

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

		Ref	Expression
	Assertion	hcauseq	⊢ ( 𝐹 ∈ Cauchy → 𝐹 : ℕ ⟶ ℋ )

Proof

Step	Hyp	Ref	Expression
1		hcau	⊢ ( 𝐹 ∈ Cauchy ↔ ( 𝐹 : ℕ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℝ⁺ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ_≥ ‘ 𝑦 ) ( norm_ℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −_ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) )
2	1	simplbi	⊢ ( 𝐹 ∈ Cauchy → 𝐹 : ℕ ⟶ ℋ )