Proving the Reciprocal Relationship of Lim Sup and Lim Inf for Bounded Sequences

[Treadstone 71]

Suppose a_n is a bounded sequence. Then prove that lim sup a_n = 1/lim inf (1/a_n).

This seems completely obvious to me, I don't know how to do this any simpler.

 

[AKG]

lim sup{a_n}
= lim_n->oosup_k>n{a_k} ... (justify this)
= lim_n->oo[1/inf_k>n{1/a_k}] ... (justify this)
= 1/[lim_n->ooinf_k>n{1/a_k}] ... (justify this)
= 1/lim inf{1/a_n} ... (justify this)

 

[Treadstone 71]

Let S={limit points of a_n}. Since a_n is strictly positive and bounded, the limit points of 1/a_n are precisely 1/s for s in S. It follows from there, doesn't it?