Nuprl Lemma : rsub-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.x[n] - y[n] = a - b)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rsub: x - 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,
rsub: x - y,
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ
Lemmas referenced :
radd-limit,
nat_wf,
rminus_wf,
rminus-limit,
converges-to_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
isectElimination,
independent_functionElimination,
functionEquality
Latex:
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,b:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.x[n] = a {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.y[n] = b {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] - y[n] = a - b)
Date html generated:
2016_05_18-AM-07_52_20
Last ObjectModification:
2015_12_28-AM-01_05_59
Theory : reals
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