4-U(n+1)=4-sqrt(12+U(n)),

using a=4-U(n)

4-sqrt(16-a)=

=4-4*sqrt(1-a/16) (1)

>=4-4(1-(1/2)a/16)

4-U(n+1)>=(1/8)(4-U(n)) - aditional rule.

Using Taulor set of sqrt:

sqrt(1-x)>=1-x and multiplying by -1, we get

-sqrt(1-x)<=-(1-x)

from (1) we get

4-U(n+1)<=(1/4)(4-U(n)).