I am trying to prove some things for a HW problem. Can you guys tell me if the following logic looks ok. Let Un and Vn be bounded sequences of real numbers. If Un<=Vn for every n , show that lim sup n--->infinity Un <= lim sup n--->infinity Vn. Here is what I wrote: Let E1 and E2 be the set of all limits of subsequences of Un and Vn respectively. E1 and E2 are bounded and nonempty becuase Un and Vn are bounded, therefore E1 and E2 have a sup in R. Since all subsequences convege to the same limit as the sequence itself, the sup of E1 is the limit of Un and the limit of E2 is the limit of Vn. Therefore since Un<=Vn for every n the lim sup of Un<=lim supVn.