Theorem hhcau

Description: The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hhlm.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
	hhlm.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	hhcau	$$⊢\mathrm{Cauchy}=\mathrm{Cau}\left(D\right)\cap \left({\mathscr{H}}^{\mathbb{N}}\right)$$

Proof

Step	Hyp	Ref	Expression
1		hhlm.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2		hhlm.2	$$⊢D=\mathrm{IndMet}\left(U\right)$$
3	1	hhnv	$$⊢U\in \mathrm{NrmCVec}$$
4	1	hhba	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(U\right)$$
5	1 3 4 2	h2hcau	$$⊢\mathrm{Cauchy}=\mathrm{Cau}\left(D\right)\cap \left({\mathscr{H}}^{\mathbb{N}}\right)$$