Nuprl Lemma : limit-shift
∀m:ℕ. ∀X:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.X[n] = a ⇒ lim n→∞.X[n + m] = a)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
nat: ℕ,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
real_wf,
converges-to_wf,
nat_plus_wf,
rless_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
rless-int,
int-to-real_wf,
rdiv_wf,
rsub_wf,
rabs_wf,
rleq_wf,
all_wf,
nat_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
introduction,
dependent_set_memberEquality,
cut,
hypothesis,
addEquality,
lemma_by_obid,
isectElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
functionEquality,
applyEquality,
inrFormation,
productElimination
Latex:
\mforall{}m:\mBbbN{}. \mforall{}X:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.X[n] = a {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.X[n + m] = a)
Date html generated:
2016_05_18-AM-07_39_21
Last ObjectModification:
2016_01_17-AM-02_04_46
Theory : reals
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