Theorem caufpm

Description: Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006) (Revised by Mario Carneiro, 24-Dec-2013)

	Ref	Expression
Assertion	caufpm	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ ( 𝑋 ↑pm ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		iscau	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( Cau ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℤ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑦 ) ) : ( ℤ≥ ‘ 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑦 ) ( ball ‘ 𝐷 ) 𝑥 ) ) ) )
2	1	simprbda	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ ( 𝑋 ↑pm ℂ ) )