1.39 Let bm,n, m, n = 1, 2,..., be a double sequence of real numbers, which for each fixed m is nondecreasing in n. Show that limn→∞ limm→∞ bm,n = limm,n→∞ inf bm,n and limm→∞ limn→∞ bm,n = limm,n→∞ sup bm,n provided the indicated limits exist (they may be infinite) and where lim inf bm,n and lim sup bm,n denote, respectively, the smallest and the largest limit points attainable by a sequence bmk ,nk , k = 1, 2,..., with mk → ∞ and nk → ∞.