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

Home Index