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%5C %5Ctextup{for some}%5C B%5Cepsilon %5Cmathbb{R}.%5C %5Ctextup{Let}%5C A%3D%5Cleft %5C{ r%5Cepsilon %5Cmathbb{R}%5Cmid S%5Ccap (r%2C%5Cinfty )%5C %5Ctextup{is finite} %5Cright %5C}%5C%5C%5C%5C A).%5C %5Ctextup{Prove that A is non-empty (It helps to recall that the empty set is a finite set)}%5C%5C%5C%5C B).%5C %5Ctextup{ Prove that A is bounded below. (hint: if}%5C r<-B%5C %5Ctextup{can}%5C r%5Cepsilon A%5C %3F)%5C%5C%5C%5C C).%5C %5Ctextup{Let}%5C %5Calpha %3D %5Ctextup{inf A. Show that}%5C %5Calpha%5C %5Ctextup{is an accumulation point for S. (hint: let}%5C %5Cepsilon >0%5C %5Ctextup{be such that}%5C (%5Calpha-%5Cepsilon %2C%5Calpha +%5Cepsilon )%5Ccap S%5C %5Ctextup{is finite%2C then find a contradiction.)}?quality=80&w=800)
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