Given the string (an), a(n+1) = a(n)(1-square root a(n)), 0

We'll re-write the "n+1"-st term of the string
(an):

a n+1 = an - an*sqrt
an

We'll subtract an both sides and we'll
get:

an+1 - an = -an*sqrt (an) <
0

Since an+1 - an, then the values of the terms of the
string (an) are decreasing as the order of the terms is increasing. So, the 1st term of
the string (an), namely a1, is the highest term.

We'll
write ak^2 = ak*ak < ak*sqrt (ak) = ak - ak+1

We'll
put k=1

a1^2 < a1 -
a2

We'll put k=2

a2^2 <
a2 -
a3

......................

ak^2
< ak - ak+1

We'll create
bn:

bn = a1^2 + ... + ak^2 < a1 - a2 + a2 -a3 + ... 
+ ak - ak+1

We'll eliminate like
terms:

bn<a1 -
ak+1

Since we have demonstrated that the string (an) is
decreasing, then a1 > ak+1, so the string (bn) is also upper limited by
a1.

The upper limit of the string (bn), if bn
= a1^2 + ... + ak^2, is a1.