Theorem 1 Txhua Cauchy sequence of real numbers converges to a limit.
Koj pom qhov txwv ntawm Cauchy li cas?
Prove: The limit of a Cauchy sequence an=limn →∞an.
Puas txhua qhov Cauchy ib ntus sib sau ua ke?
Txhua tiag Cauchy sequence is convergent. Theorem.
Puas tag nrho cov kev sib koom ua ke muaj qhov txwv?
Vim li no rau txhua qhov sib koom ua ke qhov txwv tsis pub. Piv txwv tias {an}n∈N yog convergent. Tom qab ntawd los ntawm Theorem 3.1 qhov txwv yog qhov tshwj xeeb thiab yog li peb tuaj yeem sau nws li l, hais.
Ib kab lus puas tuaj yeem hloov pauv mus rau ob qhov sib txawv?
nws txhais tau tias L1 − L2=0 ⇒ L1=L2, thiab yog li cov kab ke tsis tuaj yeem muaj ob qhov sib txawv. Rau qhov no ϵ, txij li ib qho kev sib tshuam rau L1, peb muaj tias muaj qhov ntsuas N1 kom |an −L1| N1. Nyob rau tib lub sij hawm, ib tug converges rau L2, thiab yog li ntawd muaj ib tug Performance index N2 thiaj li hais tias |an −L2| N2.
