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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
 \frac{1}{v} + \frac{1}{u} = \frac{1}{f}

Thus we can write
 \frac{1}{v}= \frac{1}{f}- \frac{1}{u}

here,  \frac{1}{f} is positive(since f is positive) and  -\frac{1}{u} is also positive(as u is negative). Since  \frac{1}{v} is sum of two positive quantities,  \frac{1}{v} always positive. So v, the image distance is always positive or the image is always virtual.
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