Suppose the numbers a0,a1,a2,....,an satisfy the following conditions:
a0= 1/2, ak+1 =ak + (1/n) ((ak)^2) for k=0,1,2,...,n-1.
Prove that (n+1)/( 2n-k+2) <ak < (n)/(2n-k )
for k=1,2,3,....,n
The statement doesn't make sense for n=0 !
We'll prove it for every natural number n>0 by induction on k.
Base case: k=1.
iff iff and this is true for every n>0.
iff and this is true for every n>0.
Base case verified.
Let for arbitrary n>0
(1)
hold for some 0<k<n . We want to show that (1) also holds for k+1, i.e. we want
(2)
to hold.
From the second inequality in (1) we have and this is strictly less than iff which is equivalent to which is true because n>0.
So the second inequality in (2) is verified.
From the first inequality in (1) we have and this is strictly greater than iff
which is equivalent to .
Now because k<n we have , no doubt.
So the first inequality in (2) is also verified and thus the induction step is completed and the proof is finished.