A very uregent question

No. The statement is wrong.

Applying the equation of motion: s = ut + (1/2)at^2
Ball A: u = 0 m/s, a = g, s = S(a), say
hence, S(a) = (1/2)gt^2

Since ball B started 0.5 s after ball A, the
S(b) = (1/2)g(t-0.5)^2

The separation between ball A and ball B = S(a) - S(b)
hence, S(a) - S(b) = (1/2)g[t^2 - (t-0.5)^2]
S(a) - S(b) = (g/2)[t^2 - t^2 + 2t - 0.25] = (g/2).[2t - 0.25]

Therefore, the separation between the two balls is not constant, but increases as time goes.

The displacement of ball A is always greater than that of ball B, but NOT by the same extent.

As time goes by, the difference in displacement of ball A and B is getting bigger and bigger.

You can prove it mathematically by using the formula:
distance = 0.5(acceleration)(time square)