Let xn ∈ R and suppose that there is an M ∈ R such that |xn| < M for n ∈ N. Prove that sn = sup{xn, xn+1,...} defines a real number for each n ∈ N and that s1 > s2 > ∙ ∙ ∙. Prove an analogous result about tn = inf{xn, xn+1,...}.