Proving the Partial Bound Question for Convergent Series An and Bn

[transgalactic]

i am given with a series called An
and series Bn which from a certain place has the same members as An?

prove or disprove that every partial bound of bn is also a partial bound of An

??

i know that if a series is converges then
lim inf An=lim sup An

is that helps?

how to prove it mathmaticly??

 

[HallsofIvy]

First, what is the definition of "partial bound"?

 

[transgalactic]

a partial bound is the bound of a subsequence