a^2 + b^2 + c^2 = 1
a, b, and c each have to be <=1 because if either was greater than 1, then a^2 + b^2 + c^2 > 1
a, b, and c each have to be >= -1 because if either was less than -1, then a^2 + b^2 + c^2 > 1
What we have now is: -1 <= a, b, c <= 1
ab, bc, ca each have to be < 1 because if either was = 1, then both parts (a and b, b and c, or c and a) would have to be 1 and 1 or -1 and -1, which would make the squared parts (a^2 + b^2, b^2 + c^2, or c^2 + a^2) > 1
So the <=1 part of 1/2 <= ab + bc + ca <= 1 is incorrect.
Now consider the case of a=1, b=0, c=0. That satisfies a^2 + b^2 + c^2 = 1. But does it satisfy 1/2 <= ab + bc + ca ?
ab + bc + ca
1(0) + 0(0) + 0(0 = 0
0 is not greater than or equal to 1/2.
So the 1/2 <= part of 1/2 <= ab + bc + ca <= 1 is incorrect.
Answer: a^2 + b^2 + c^2 = 1 does not mean 1/2 <= ab + bc + ca <= 1