Metamath Proof Explorer


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 : F Cauchy x + y z y norm F y - F z < x

Proof

Step Hyp Ref Expression
1 hcau F Cauchy F : x + y z y norm F y - F z < x
2 1 baib F : F Cauchy x + y z y norm F y - F z < x