I was reading The Accuracy of Approximate Calculations in Numerical Analysis by K. Mukherjee; there the author mentioned a note after introducing absolute error to be followed:
Note: The absolute error in a numerical computation correct to $n$ significant figures cannot be greater than half a unit in $n$-th place.
Why is it so? Why cannot the absolute error be greater than "half a unit in $n$-th place"?
The author didn't provide any derivation or how he concluded this; is there any proof of this?
This is true only if default rounding mode (i.e. round to a nearest value) is used and the result do not underflow or overflow.
However according to IEEE-754 standard different rounding modes are possible, for example rounding toward $+\infty$ (ceiling) or rounding toward $-\infty$ (flooring). With these non default rounding modes it is possible, that roundoff error can be larger than half of and unit in the last place.
In general, if a number do not underflow or overflow, then roundoff error cannot be larger than unit in the last place. If a number underflows or overflows, then roundoff error can be much larger.