Proving properties of lim sup. Good attempt at proof

[tamintl]

[/itex]

Homework Statement

Is the following statement true or false: 'if (c_n) and (d_n) are bounded sequences of positive real numbers then:

lim sup (c_nd_n) = (lim sup c_n)(lim sup d_n)

Homework Equations

The Attempt at a Solution

for all n in the positive reals. c_n and d_n are bounded.

Since c_n and d_n are bounded we know they converge.

Hence, by the theorem lim(c_nd_n)=lim c_nlim d_n we can say that lim(c_n)=lim sup (c_n) and lim(d_n)=lim sup (d_n)

Hence, lim sup (c_nd_n) = (lim sup c_n)(lim sup d_n)

HENCE, TRUE!

******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******

 

[gopher_p]

******I am not sure about line 2 where I say bounded => converge. I know that you can say that converge means bounded but I don't know if I can do the reverse.*******

You can't. You can say that a bounded monotone sequence converges and that a bounded sequence has a convergent subsequence. But a bounded sequence need not converge. You should be able to come up with a pretty simple example of a bounded sequence that does not converge (at least now that you know there is one). In the process of doing so, you may stumble across the answer to your problem.