Question

Give examples, if possible, of the following.

i) A set with a supremum but no maximum.

ii) A decreasing sequence $(a_n)_{n=1}^{\infty}$ so that $inf\left \{ a_n|n\in\mathbb{N} \right \}$ does not exist

iii) An increasing sequence $(a_n)_{n=1}^{\infty}$ so that $inf\left \{ a_n|n\in\mathbb{N} \right \}$ does not exist.

Answer 1

we know suppemim ol ce set A is, s such that EXEA and if t such that ast,VICEA, them. sst. Also, maximcem of set A is s such that for all XEA, als and SEA How consider A = { x GIR (x?< v), 2} Then, Sup A = 2 r? =) to sc<12KX EA} But Maximum of A does not exists of 2 does not belong to (A I now, consider the sequence, Bon is a deoseasing sequence an = thers, {an as, - n - Cnt) But an > anti Int inf {an / nein} does not exist ad as {anina is dexreasing a sequensce which is not bounded below so an = lim on n-) no of inffannen does not exists.

so nu since lan} is ineseasing sequence Wi>2 A so, a, is a, is lower boundo of am] Suppose if I to such that ti's intiimum of { an} than , t sai vi z 1 locer bound t sai al is greatest lower bound. a, is infimum of {an}. . Given example does not eseists.