`∀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` radd: `a + b` 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` uall: `∀[x:A]. B[x]` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` prop: `ℙ` sq_exists: `∃x:{A| B[x]}` nat: `ℕ` guard: `{T}` 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` uiff: `uiff(P;Q)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rneq: `x ≠ y` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rev_uimplies: `rev_uimplies(P;Q)` rge: `x ≥ y` rsub: `x - y`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution cut hypothesis dependent_functionElimination thin introduction extract_by_obid isectElimination dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality hypothesisEquality baseClosed setElimination rename dependent_set_memberFormation equalityTransitivity equalitySymmetry applyLambdaEquality unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination because_Cache productElimination functionEquality applyEquality functionExtensionality inrFormation minusEquality multiplyEquality addEquality

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: 2017_10_03-AM-09_04_53
Last ObjectModification: 2017_07_28-AM-07_41_12

Theory : reals

Home Index