Could you help me prove these inequalities about lim inf and lim sup of a sequence?

Question

Let x_n be a sequence of real numbers and w_n a sequence of positive weights. Let s_n = (Sum(i= 1,n ) w_n x_n)/(Sum (i=1, n) w_n) be the sequence of the weighted means of x_n with respect to w_n. Show that, if Sum (w_n) diverges, then in the extended real system, the following inequalities hold:

lim inf x_n <= lim inf s_n <= lim sup s_n <= lim sup x_n

(of course, the middle inequality is always true, and it suffices to show either the left or the right inequality).

Show these inequalities don't need to be true if Sum (w_n) converges.

Thank you for any help

Answer 1

I'm in a hurry now, so I'll give you some hints. Tomorrow, if no one has answered your question, I'll finish up my proof. But I think with such hints you can do it yourself.

Based on what you said, let's consider the leftmost inequality. If lim inf x_n = -oo, then we have nothing to prove.So, suppose lim inf x_n > -oo. For every v < lim inf x_n, there is an integer k such that n > k => x_n > v. In the definition of s_n, split the numerator into 2 sums, the first from 1 to k, the second from k to n, n> k. Plug in v for i >k. You get a sequence, call it t_n, and s_n dominates t_n for n > k. With some algebra you'll see t_n --> v. Then,

lim inf s_n >= lim inf t_n = lim t_n = v. Since this is true for every v < lim inf x_n, you get something interesting.

For the second part, observe that, if Sum w_n converges, then it's possible that x_n converges to some x and s_n converges to something else. In such cases, it's impossible to satisfy such inequalities.

Hope this helps.