Nuprl Lemma : continuous-limit
∀I:Interval. ∀f:I ⟶ℝ. ∀y:ℝ. ∀x:ℕ ⟶ ℝ.
(f(x) continuous for x ∈ I ⇒ lim n→∞.x[n] = y ⇒ (y ∈ I) ⇒ (∀n:ℕ. (x[n] ∈ I)) ⇒ lim n→∞.f(x[n]) = f(y))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
r-ap: f(x),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
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]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
continuous: f[x] continuous for x ∈ I,
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
sq_stable: SqStable(P),
nat: ℕ,
rneq: x ≠ y,
guard: {T},
rev_implies: P ⇐ Q,
rless: x < y,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
real: ℝ,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
interval: Interval,
i-approx: i-approx(I;n),
i-member: r ∈ I,
rccint: [l, u],
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
i-member-iff,
i-approx-monotonic,
mul_nat_plus,
less_than_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
i-approx-compact,
icompact_wf,
i-approx_wf,
nat_plus_wf,
all_wf,
nat_wf,
i-member_wf,
converges-to_wf,
continuous_wf,
r-ap_wf,
sq_stable__i-member,
real_wf,
rfun_wf,
interval_wf,
sq_stable__rless,
int-to-real_wf,
le_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_properties,
decidable__lt,
rless_wf,
small-reciprocal-real,
rless_transitivity2,
rleq_weakening_rless,
imax_wf,
imax_nat,
sq_stable__less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
imax_lb,
rabs-difference-bound-rleq,
radd_wf,
multiply_nat_plus,
req-int-fractions,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
req_functionality,
radd-int-fractions,
req_weakening,
rleq-int-fractions,
req-int-fractions2,
rleq_functionality,
req_wf,
true_wf,
rmul_wf,
rminus_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rleq_transitivity,
rsub_functionality_wrt_rleq,
rleq_weakening,
req_inversion,
radd_functionality,
uiff_transitivity,
rsub_functionality,
req_transitivity,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
radd-assoc,
radd_comm,
radd-ac,
rminus-as-rmul,
rmul-one-both,
radd_functionality_wrt_rleq,
radd-preserves-rleq,
rminus-radd,
radd-rminus-assoc,
rmul-zero-both,
radd-zero-both,
rleq-int,
uimplies_transitivity,
subtract_wf,
itermMinus_wf,
itermSubtract_wf,
int_term_value_minus_lemma,
int_term_value_subtract_lemma,
rsub-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
addLevel,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_isectElimination,
setElimination,
rename,
multiplyEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
levelHypothesis,
applyEquality,
functionExtensionality,
imageElimination,
setEquality,
functionEquality,
inrFormation,
dependent_set_memberFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
addEquality,
productEquality,
minusEquality
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}y:\mBbbR{}. \mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
(f(x) continuous for x \mmember{} I
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] = y
{}\mRightarrow{} (y \mmember{} I)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}. (x[n] \mmember{} I))
{}\mRightarrow{} lim n\mrightarrow{}\minfty{}.f(x[n]) = f(y))
Date html generated:
2017_10_03-AM-10_18_54
Last ObjectModification:
2017_07_28-AM-08_06_13
Theory : reals
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