Let (X , d) be a compact metric space and fn : X > R be a sequence of continuous real functions. Assume that Vn Z 1, Va: : fn(:r) 2 fn+1(a:) and that their pointwise limit limnnc,o f" =: f is itself continuous. Show that then f ) f uniformly. Hint: Show rst that without loss of generality f can be assumed to be identically 0. Then one possibility to proceed is as follows. Since X is compact and fn is continuous it takes on its maximum, choose such a point say 2:", and then we know that for all y E X, 0 S fn(y) S fn(a:n). Show next that the sequence fats") is monotone decreasing. Since it is bounded from below by 0 it converges. Show that if its limit is 0 then fn ) 0 uniformly. Finally it remains to show that 7 := limme fn(:cn) = 0. Assume that this limit is strictly positive, (7 > O) and nd a contradiction. For this consider the sequence xn and use (seq.) compactness. Find an example where X is not compact, fn are still continuous and they monotonically decrease to the limit f pointwise, (f \\=: f) yet the convergence is not uniform even though f itself is continuous. Recall that a series . > an is called convergent iff the sequence of their partial sums S, := k=1...n an is convergent: Sn + s, and in this case the infinite sum _, an is defined as s