Database
COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
Cauchy sequences and completeness axiom
Cauchy sequences and limits
seq1hcau
Metamath Proof Explorer
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 norm ℎ ⁡ 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 norm ℎ ⁡ F ⁡ y - ℎ F ⁡ z < x
2
1
baib
⊢ F : ℕ ⟶ ℋ → F ∈ Cauchy ↔ ∀ x ∈ ℝ + ∃ y ∈ ℕ ∀ z ∈ ℤ ≥ y norm ℎ ⁡ F ⁡ y - ℎ F ⁡ z < x