Nuprl Lemma : function-limit
∀I:Interval. ∀f:I ⟶ℝ. ∀y:ℝ. ∀x:ℕ ⟶ ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f(x) = f(y))))
⇒ lim n→∞.x[n] = y
⇒ (y ∈ I)
⇒ (∀n:ℕ. (x[n] ∈ I))
⇒ lim n→∞.f(x[n]) = f(y))
Proof
Definitions occuring in Statement :
r-ap: f(x),
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
converges-to: lim n→∞.x[n] = y,
req: x = y,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
so_apply: x[s],
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
rfun: I ⟶ℝ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
interval_wf,
rfun_wf,
nat_wf,
req_wf,
all_wf,
i-member_wf,
real_wf,
sq_stable__i-member,
r-ap_wf,
function-is-continuous,
continuous-limit
Rules used in proof :
functionEquality,
setEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
because_Cache,
rename,
setElimination,
isectElimination,
lambdaEquality,
sqequalRule,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}y:\mBbbR{}. \mforall{}x:\mBbbN{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (f(x) = f(y))))
{}\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_19_08
Last ObjectModification:
2017_07_31-AM-11_47_30
Theory : reals
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