How to prove that this series bounded and monotonic

[transgalactic]

Xn=(1-1/2)(1-1/4)..(1-(1/(2^n))

i tried to prove that its monotonic
by :
1-1/(2^n) = (2^n-1)/2^n

2^n -1 <2^n
obviously its correct
the numerator of each object is smaller then the denominator.

what now??

and how to prove that its bounded?

 

[Office_Shredder]

You need to show that $$x_n > x_{n+1}$$ for all n given that $$x_n(1-2^{-n}) = x_{n+1}$$

It should be fairly easy from here

 

[transgalactic]

that proves that its monotonic
how to prove that its bounded?

 

[Office_Shredder]

Is it increasing or decreasing?

 

[transgalactic]

each next member is bigger then the previous one
so its increasing

1-1/2 1-1/4 1-1/8 etc..

 

[Office_Shredder]

That's now what the sequence is. The sequence is

1/2, 1/2*3/4, 1/2*3/4*7/8 etc.

you should be able to see this from how x_n is defined.

 

[transgalactic]

ok so it getting smaller and smaller
how to prove that its bounded?

 

[HallsofIvy]

There's a pretty obvious lower bound. And since it is decreasing, isn't x₁= 1/2 an upper bound?