For a convex mirror, focal length(f) is always positive and object distance (u) is always negative according to the convention.
We have to show that the image formed is virtual. A virtual image means image is formed behind the mirror or image distance(v) is always positive.
Mirror formula is given as
Thus we can write
is positive(since f is positive) and
is also positive(as u is negative). Since
is sum of two positive quantities,
always positive. So v, the image distance is always positive or the image is always virtual.