Average of two estimates

Hi, I'm currently studying the Babylonian method of calculating square roots and trying to understand why the average of our two guesses (k + S/k) is always a better approximation than either k or S/k.

From what I can see, if one estimate is above the target and the other below, as long as the larger error is less than x3 the lower error away from the target, the average is guaranteed to be a better approximation.

For example, if the target value is 50 and the low and high estimates are 25 and 75 respectively, then the average is precisely the target value, 50. If on the other hand, the lower value is 25 and the higher value is (target_value + 3 * (target_value - lower_value)) = (50 + 3*25) = 125, then the average is 75 which is the same absolute error as the lower value.

Therefore, is it safe to assume that as long as the errors are within this range, then the average will always be closer than either estimates?

I suppose if this is the case, then as long as we can prove that this holds true for (k+S/k), then I can be happy that this method will always work.

I realise there are far more complicate methods to prove/disprove the above (I know little of limits & calculus), but I am a mature student currently studying elementary math so, please, be kind and keep it simple