Show that if x_n > 3^{0.5} then sequence x_{n+1} > 3^{0.5}

Hi, I have the following problem which I don't know how to solve... I know how to do part C, but I don't know parts A and B. I will have enough to guide myself if you help me with part A though.

Thanks a lot!

The whole problem is:

"Let for . Define a sequence of real numbers by

for , .

A) Show that if , then .

B) Show that if , then . (Hint: show .)

C) Conclude that the sequence converges."

Re: Show that if x_n > 3^{0.5} then sequence x_{n+1} > 3^{0.5}

Quote:

Originally Posted by

**buenogilabert** Hi, I have the following problem which I don't know how to solve... I know how to do part C, but I don't know parts A and B. I will have enough to guide myself if you help me with part A though.

Thanks a lot!

The whole problem is:

"Let

for

. Define a sequence of real numbers

by

for

,

.

A) Show that if

, then

.

B) Show that if

, then

. (Hint: show

.)

C) Conclude that the sequence

converges."

A)

(AM-GM inequality, which holds for )

So, if , then .

Equality holds only when

So, if and , then .

In particular, if , then .