Proof of limsup(anbn)<=limsup(an)limsup(bn) and Example

[stukbv]

Homework Statement

if (an) and (bn) are bounded positive sequences prove that
limsup(anbn)<= limsup(an)limsup(bn)
and give an example to show there is not equality in general

 

[HallsofIvy]

If you do not show at least some attempt to do this problem yourself, this thread will be deleted.

 

[stukbv]

all i know is, if we let l = limsupan and m = limsupbn ,
then we can say that there exists an n such that
l-e < an < l+e
m-e<bn < l + e

so |anbn| < (l+e)(m+e)
...

 

[micromass]

Hi stukbv!

Can you first show that

$$\sup{(a_nb_n)}\leq \sup{a_n}\sup{b_n}$$