Let n from 1 to infinity and n from 1 to infinity be bounded sequences of real numbers. Prove that <= .
n N
Conclude that <= + . Formulate analogus results for inf and liminf (no need to prove those).
The example i could think of is and . 0<2. [tex] How to prove it in general?
I think it so hard for directly prove.
My definition is
Proof. Let . Assume , and Then there exists such that and Choose For all , we have and so or . Since is bounded, the Bolzano- Weierstrass implies that there is a convergent subsequence of such that , with . Thus, for all , take limit as we get and so Hence