the question in this link:
http://img372.imageshack.us/img372/7929/76106583tn6.gif

this question is obvious
its common sense

if a(n) converges to a

then if we take the function who takes the biggest member
of course it will pick the closest to "a"

it take the largest member from a1 to an

for example:
a1=6 a2=5 a3=6 a4=1 a5=8 a6=7

b1=6 b2=6 b3=6 b4=6 b5=8 b6=8

i dont know how to transform these word into math

??

the definition of convergens states that
if a sequence is bounded and monotonic then it converges

An->a
Bn=max{till An }