True/false: proportional controller makes error zero

False. A proportional controller only reduces the steady-state error; it cannot make it zero (unless the plant already contains an integrator).

Justification. With $G_c(s) = K_p$ and a type-0 plant $G(s)$ in a unity feedback loop, the step error is

$$e_{ss} = \lim_{s\to 0} \frac{s\cdot\frac{1}{s}}{1 + K_p G(s)} = \frac{1}{1 + K_p G(0)}$$

Example: $G(s) = \dfrac{1}{s+1}$, so $G(0) = 1$.

The error falls as $K_p$ rises but becomes zero only for $K_p \to \infty$, which is not possible. Very high gain also makes higher-order systems oscillatory or unstable and saturates the actuator.

A P controller adds no pole at the origin, so it does not change the system type. Zero steady-state error needs integral action (PI or PID controller), which adds a pole at $s = 0$.