Q: Let be a bounded (not necessarily convergent) sequence, and assume . Show that . Why are we not allowed to use the Algebraic Limit Theorem to prove this?

For the last sentence is seems clear that we cannot use the ALT to prove this, because we need both limits to converge to use the ALT.

For the proof I have written out the definitions that pertain to statement; such as, the definition of a bounded sequence and convergence. I have also drawn a picture and I think I understand why this statement is true, but I am having trouble formalizing a proof.

Any help would be great.