Ok, we want to compute the lower fixed point of for, as usual, .

We do that by iterating:

while iterating we can not directly compute the difference but we can compute and we ask our self how close must come to 1 such that

First we translate the difference into a quotient:

is decreasing. Now lets compute

To make an estimate we want to know whether , is decreasing for . To decide this we differentiate:

for

This is true for because then .

So we know that is strictly increasing for small enough . So we get

iff

for , equivalently:

iff

iff

The right side can be further simplified:

The condition is satisfied for , i.e. on the right side is a constant . So during the iteration we can decrease according to without fear that becomes to small and the iteration does not stop.

Proposition. Let , let be the lower fixed point of , let and then for each :

iff .

We do that by iterating:

while iterating we can not directly compute the difference but we can compute and we ask our self how close must come to 1 such that

First we translate the difference into a quotient:

is decreasing. Now lets compute

To make an estimate we want to know whether , is decreasing for . To decide this we differentiate:

for

This is true for because then .

So we know that is strictly increasing for small enough . So we get

iff

for , equivalently:

iff

iff

The right side can be further simplified:

The condition is satisfied for , i.e. on the right side is a constant . So during the iteration we can decrease according to without fear that becomes to small and the iteration does not stop.

Proposition. Let , let be the lower fixed point of , let and then for each :

iff .