Nuprl Lemma : mconverges-implies-mcauchy
∀[X:Type]. ∀[d:metric(X)]. ∀[x:ℕ ⟶ X]. (x[n]↓ as n→∞ ⇒ mcauchy(d;n.x[n]))
Proof
Definitions occuring in Statement :
mconverges: x[n]↓ as n→∞,
mcauchy: mcauchy(d;n.x[n]),
metric: metric(X),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
mconverges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
mcauchy: mcauchy(d;n.x[n]),
all: ∀x:A. B[x],
mconverges-to: lim n→∞.x[n] = y,
member: t ∈ T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
and: P ∧ Q,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
rev_uimplies: rev_uimplies(P;Q),
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
ge: i ≥ j ,
rge: x ≥ y,
uiff: uiff(P;Q),
metric: metric(X),
so_lambda: λ2x.t[x],
req_int_terms: t1 ≡ t2
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x:\mBbbN{} {}\mrightarrow{} X]. (x[n]\mdownarrow{} as n\mrightarrow{}\minfty{} {}\mRightarrow{} mcauchy(d;n.x[n]))
Date html generated:
2020_05_20-AM-11_56_31
Last ObjectModification:
2019_12_14-PM-04_48_58
Theory : reals
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