Nuprl Lemma : unique-limit
∀[x:ℕ ⟶ ℝ]. ∀[y1,y2:ℝ]. (y1 = y2) supposing (lim n→∞.x[n] = y2 and lim n→∞.x[n] = y1)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
converges-to: lim n→∞.x[n] = y,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
sq_exists: ∃x:{A| B[x]},
so_apply: x[s],
nat: ℕ,
implies: P ⇒ Q,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
infinitesmal-difference,
mul_nat_plus,
less_than_wf,
sq_stable__rleq,
rabs_wf,
rsub_wf,
nat_wf,
imax_wf,
imax_nat,
nat_properties,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
intformless_wf,
itermMultiply_wf,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
rless_wf,
imax_ub,
less_than'_wf,
nat_plus_wf,
req_witness,
converges-to_wf,
real_wf,
radd_wf,
rleq_wf,
rminus_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
r-triangle-inequality,
uiff_transitivity,
rleq_functionality,
req_weakening,
rabs_functionality,
req_inversion,
radd-assoc,
req_transitivity,
radd-ac,
radd_functionality,
radd-rminus-assoc,
radd_functionality_wrt_rleq,
rabs-difference-symmetry,
rmul_wf,
rleq-int-fractions,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-int-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
because_Cache,
setElimination,
rename,
applyEquality,
functionExtensionality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
multiplyEquality,
inrFormation,
imageElimination,
inlFormation,
independent_pairEquality,
minusEquality,
axiomEquality,
functionEquality,
addEquality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[y1,y2:\mBbbR{}]. (y1 = y2) supposing (lim n\mrightarrow{}\minfty{}.x[n] = y2 and lim n\mrightarrow{}\minfty{}.x[n] = y1)
Date html generated:
2017_10_03-AM-08_52_28
Last ObjectModification:
2017_07_28-AM-07_35_11
Theory : reals
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