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.