Let (xn) be a bounded sequence, and for each n ∈ n let sn : = sup{xk : k ≥ n} and tn : = inf{xk : k ≥ n}. prove that (sn) and (tn) are monotone and convergent. also prove that if lim sn = lim tn, then (xn) is convergent. [one calls limsn the limit superior of (xn), lim sup xn, and lim tn the limit inferior of (xn), lim inf xn.]