Ok, sorry. It's a proof for the Theorem:
"For each positive real number a and each integer n > 1, there is a unique real number b such that
In the special case
The proof begins with
etc.
to give decimal b^2 = (1.414...)^2 = 2
Then the method uses a table
b b b^2 1 1 1 1.4 1.4 1.96 1.41 1.41 1.9881 1/414 1/414 1.999396 etc.
to prove so.
Then to prove the least upper bound of the set of numbesr in the third column of the table is 2, we check that M=2 is an upper bound of E, which follows from the inequalities at the top. Then to show if M' < 2 there is a number in E that is greater then M'.
We put
Then we have
so