Proof by induction of a sequence.

[missavvy]

Homework Statement

Given a sequence of rational numbers, defined inductively as follows:

s₁ = 1, s_n+1 = s_n/2 + 1/s_n, n>=1

prove that 1<=s_n<=2 forall n>=1

Homework Equations

The Attempt at a Solution

I've got the solution to this but I don't understand a certain part, I was hoping someone could explain it to me?

So Let P(n) be the statement 1<=s_n<=2
P(1) is satisfied.
Suppose P(n) holds.
Then s_n+1 = s_n/2 + 1/s_n >= 1/2 + 1/2 = 1, because s_n/2>=1/2 and 1/s_n<=1/2

The bold part is what i don't understand. Wouldn't that imply that s_n >= 2?
Am I mistaken or is there maybe an error in this solution?

Thanks in advance.

 

[Office_Shredder]

The inequality is written backwards. It should read $$\frac{1}{s_n} \geq \frac{1}{2}$$, which of course is what you need to write the >= earlier in that line

 

[missavvy]

thanks I thought so... But then this solution continues as follows...

Moreover, S_n+1 = s_n/2 + 1/s_n <= 2/2 + 1/2 <= 3/2 <= 2

I'm unsure about this, because then wouldn't you need 1/s_n <= 1/2 for this step?

____________________


Ok forgetting about that solution, if I were to say:

S_n+1 = S_n/2 + 1/S_n >= 1/2 + 1/2 = 1
and
S_n+1 = s_n/2 + 1/S_n <= 1 + 1 = 2

Is this right and proves that P(n+1) holds?

 

[╔(σ_σ)╝]

You can use AM-GM inequality.