Nuprl Lemma : converges-implies-bounded
∀x:ℕ ⟶ ℝ. (x[n]↓ as n→∞ ⇒ bounded-sequence(n.x[n]))
Proof
Definitions occuring in Statement :
bounded-sequence: bounded-sequence(n.x[n]),
converges: x[n]↓ as n→∞,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x.t[x],
so_apply: x[s],
bounded-sequence: bounded-sequence(n.x[n]),
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtype_rel: A ⊆r B,
rge: x ≥ y,
rsub: x - y,
uiff: uiff(P;Q),
sq_type: SQType(T)
Lemmas referenced :
less_than_wf,
converges_wf,
nat_wf,
real_wf,
radd_wf,
rabs_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
less_than'_wf,
rsub_wf,
nat_plus_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
nat_plus_wf,
all_wf,
rleq_wf,
r-triangle-inequality,
equal_wf,
rminus_wf,
squash_wf,
true_wf,
radd_comm_eq,
iff_weakening_equal,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
radd-ac,
radd_comm,
radd_functionality,
radd-rminus-both,
req_weakening,
radd-zero-both,
bounded-sequence_wf,
intformless_wf,
int_formula_prop_less_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_wf,
primrec-wf2,
add-zero,
rmax_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
zero-add,
rleq-rmax,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
imageMemberEquality,
hypothesisEquality,
baseClosed,
extract_by_obid,
isectElimination,
setElimination,
rename,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
dependent_pairFormation,
because_Cache,
independent_isectElimination,
inrFormation,
independent_functionElimination,
independent_pairEquality,
voidElimination,
addEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
instantiate,
cumulativity
Latex:
\mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}. (x[n]\mdownarrow{} as n\mrightarrow{}\minfty{} {}\mRightarrow{} bounded-sequence(n.x[n]))
Date html generated:
2017_10_03-AM-08_52_41
Last ObjectModification:
2017_07_28-AM-07_35_18
Theory : reals
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