The statement "objects are closer than they appear" in a car's convex mirror is not technically correct. Prove this using maths.
Isn't it correct? A convex mirror gives you a wider field of view to overcome the passenger side blind spot. The wider the view, the smaller the object, giving the false impression that the objects in question are more distant than they are.

Now for some formulae
1/-f=(1/do)+(1/di)
do is distance from the object to the mirror, and di is the distance from the image to the mirror. f is the focal point
hi/ho=di/do=m
hi is height of the image and ho is height of the object. m is magnification.

Let's say that ho=1.5, f=-2.5 (for a convex mirror the focal point is negative), and do=2.5
1/-2.5=(1/2.5)+1/di
(1/-2.5)-(1/2.5)=(1/di)
(1/-2.5)-(1/2.5)=0.8=
(1/0.8)=-1.25=di
So di is equal to -1.25

Moving on....
(hi/1.5)=(1.25/2.5)
(1.25/2.5)=0.5
1.5*0.5=0.75=hi
0.75/1.5=0.5
So the image is half its size! If we were to view this image in the mirror, it would appear to be farther away because we intrinsically know that an object will appear smaller the farther away it is.

I would have to agree with you, although I guess it depends on how you define "appear", as you could judge the distance in multiple ways (size, size compared to other objects to which you know the distance, etc). But if we go with the distance to the virtual image (where your eyes focus), it certainly is closer than it appears.