Final value theorem for Z-transform

Final value theorem: if X(z) = Z[x(k)] and (z − 1)X(z) has no poles on or outside the unit circle, then x(∞) = lim_{z→1} (z − 1) X(z).

Take x(kT) = 1 − e^(−akT), a > 0, so e^(−aT) < 1. Then

X(z) = z/(z − 1) − z/(z − e^(−aT))
(z − 1) X(z) = z − z(z − 1)/(z − e^(−aT))

The only pole of (z − 1)X(z) is at z = e^(−aT), which lies inside the unit circle, so the theorem applies. Putting z = 1:

x(∞) = 1 − 1·0/(1 − e^(−aT)) = 1

Answer: x(∞) = 1 (agrees with 1 − e^(−a·∞) = 1).