Proving 4-U_{n+1}≤(1/4)(4-U_{n})

[mtayab1994]

Homework Statement

$$U_{n+1}=\sqrt{12+U_{n}}$$ V0=0

Homework Equations

1) prove that Un<4
2) prove that $$4-U_{n+1}\leq\frac{1}{4}(4-U_{n})$$
3) conclude that 4-Un<(1/4)^(n-1)

The Attempt at a Solution

1- for n=0 0<4
assume Un<4 for some n in N and prove that Un+1<4

√(12+Un)<4 then square both sides and we get:

12+Un<16 then Un<4
so for every n in N: Un<4

2) I don't know what to do any help?

 

[obafgkmrns]

2) I don't know what to do any help?

Have you tried substituting √(12+Un) for U(n+1) in $$4-U_{n+1}\leq\frac{1}{4}(4-U_{n})$$ and then evaluating the resulting expression for the largest possible value of U(n+1)?

 

[mtayab1994]

2) I don't know what to do any help?

Have you tried substituting √(12+Un) for U(n+1) in $$4-U_{n+1}\leq\frac{1}{4}(4-U_{n})$$ and then evaluating the resulting expression for the largest possible value of U(n+1)?

Yea i just did that thanks for your help