Consider the sequences defined as followsan=-1^nbn=1ncn=n^2dn=6n+47n-3a For each sequence give an example of a monotone subse-quenceb For each sequence give its set of subsequential limitsc For each s

(a)

• For an, the subsequences (-1)^{2n} and (-1)^{2n+1} are monotone.
• For bn, the subsequence 1/n is monotone.
• For cn, any subsequence of the form n^2 is monotone.
• For dn, any subsequence that is either increasing or decreasing is monotone. For example, the subsequence (6n+4)/(7n-3) when n is odd is decreasing, and the subsequence (6n+4)/(7n-3) when n is even is increasing.

(b)

• The set of subsequential limits of an is {-1, 1}.
• The set of subsequential limits of bn is {0}.
• The set of subsequential limits of cn is {+infinity}.
• The set of subsequential limits of dn is {6/7}.

(c)

• The lim sup of an is 1, and the lim inf is -1.
• The lim sup of bn is 0, and the lim inf is 0.
• The lim sup of cn is +infinity, and the lim inf is +infinity.
• The lim sup of dn is 6/7, and the lim inf is 6/7.

(d)

• The sequence an diverges to both +infinity and -infinity.
• The sequence bn converges to 0.
• The sequence cn diverges to +infinity.
• The sequence dn converges to 6/7.

(e)

• The sequences bn, cn, and dn are bounded.
• The sequence an is unbounded