From 0<|x_n - x|<|x|/20<|xn−x|<|x|2 we have equivalently
epsilon_1^2=sqrt((x_n-x)^2)ε21=√(xn−x)2
sqrt((x_n-x)^2)+epsilon_2^2 = sqrt(((x)/2)^2)√(xn−x)2+ε22=√(x2)2
with {epsilon_1,epsilon_2} ne 0{ε1,ε2}≠0
then
(sqrt((x_n-x)^2)+epsilon_2^2)^2 = ((x)/2)^2(√(xn−x)2+ε22)2=(x2)2 or
(x_n-x)^2+2epsilon_2^2 sqrt((x_n-x)^2)+epsilon_2^4=x^2/4(xn−x)2+2ε22√(xn−x)2+ε42=x24
and again
(2epsilon_2^2 sqrt((x_n-x)^2))^2=(x^2/4-(x_n-x)^2-epsilon_2^4)^2(2ε22√(xn−x)2)2=(x24−(xn−x)2−ε42)2
or factoring
-1/16 (2 epsilon_2^2 + x - 2 x_n) (2 epsilon_2^2 + 3 x - 2 x_n) (2 epsilon_2^2 - 3 x +
2 x_n) (2 epsilon_2^2 - x + 2 x_n)=0−116(2ε22+x−2xn)(2ε22+3x−2xn)(2ε22−3x+2xn)(2ε22−x+2xn)=0
or
{(2 epsilon_2^2 + x - 2 x_n=0), (2 epsilon_2^2 + 3 x - 2 x_n=0), (2 epsilon_2^2 - 3 x + 2 x_n=0), (2 epsilon_2^2 -
x + 2 x_n=0):}
or equivalently
{(x/2 - x_n < 0), (3/2 x - x_n < 0), ( - 3/2 x + x_n < 0), (-
x/2 + x_n < 0):}
or equivalently
x/2< x_n and x_n < x/2
3/2x < x_n and x_n < 3/2 x
Those results are contradictory.
