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Let(bn)be a bounded sequence of positive real number such that for any bounded positive sequence of real number we have: limsup(an bn)=...

Let(bn)be a bounded sequence of positive real number such that for any bounded positive sequence of real number we have:
limsup(an bn)= limsup(an)limsup(bn). Prove (bn) is convergent sequence
or on the other hand: I am having a hard time to solve this problem please help.
lim n->infinity bn/an=0 Here how can i find a sequence an and a sequence bn without using Ratio Test. The Limit above is true and the Ratio Test gives no conclusion about the convergence of either "sigma an or sigma bn" even though sigma bn is much smaller than the terms of sigma bn.

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