us ual ) ▁limit ▁in ▁. ▁ ▁In ▁other ▁words , ▁a ▁Ban ach ▁limit ▁extends ▁the ▁usual ▁limits , ▁is ▁linear , ▁shift - in variant ▁and ▁positive . ▁ ▁However , ▁there ▁exist ▁sequences ▁for ▁which ▁the ▁values ▁of ▁two ▁Ban ach ▁limits ▁do ▁not ▁agree . ▁We ▁say ▁that ▁the ▁Ban ach ▁limit ▁is ▁not ▁uniqu ely ▁determined ▁in ▁this ▁case . ▁ ▁As ▁a ▁consequence ▁of ▁the ▁above ▁properties , ▁a ▁real - valu ed ▁Ban ach ▁limit ▁also ▁satisfies : ▁▁▁▁ ▁The ▁existence ▁of ▁Ban ach ▁limits ▁is ▁usually ▁proved ▁using ▁the ▁H ahn – B an ach ▁theorem ▁( anal yst ' s ▁approach ), ▁or ▁using ▁ult ra filters ▁( this ▁approach ▁is ▁more ▁frequent ▁in ▁set - the oret ical ▁expos itions ). ▁These ▁proofs ▁necessarily ▁use ▁the ▁A xi om ▁of ▁choice ▁( so ▁called ▁non - effect ive ▁proof ). ▁ ▁Al most ▁convergence ▁There ▁are ▁non - con ver gent ▁sequences ▁which ▁have ▁a ▁uniqu ely ▁determined ▁Ban ach ▁limit . ▁ ▁For ▁example , ▁if ▁, ▁then ▁ ▁is ▁a ▁constant ▁sequence , ▁and ▁ ▁holds . ▁ ▁Thus , ▁for ▁any ▁Ban ach ▁limit , ▁this ▁sequence ▁has ▁limit ▁. ▁ ▁A ▁bounded ▁sequence ▁ ▁with ▁the ▁property , ▁that ▁for ▁every ▁Ban ach ▁limit ▁ ▁the ▁value ▁ ▁is ▁the ▁same , ▁is ▁called ▁almost ▁conver gent . ▁ ▁Ban ach ▁spaces ▁Given ▁a ▁conver gent ▁sequence ▁ ▁in ▁, ▁the ▁ordinary ▁limit ▁of ▁ ▁does ▁not ▁arise ▁from