In this situation, is an endless sequence of images (theoretically) formed?

I was dealing with an optics problem, where an object is placed at $u= -60$ cm in front of a biconvex lens of radius of curvature $R=30$ cm and refractive index $\mu=1.5$ . Simple calculations show that an image (real and inverted) is formed at $v=60$ cm .

This isn't what bothers me.

The curved surface of the lens facing the object also acts as a convex mirror of focal length $f=15$ cm, and produces a virtual, erect image of the object at $v_1=10$ cm.

If this virtual image (let's call it $I_1$) were to act as a virtual object at $u_2=10$ cm for the lens, it would produce another virtual image ($I_2$) at $v_2=7.5$ cm.

This process could go on, with each virtual image at $v_n$ acting as a virtual object at $u_{n+1}$ for the lens to give another image at $v_{n+1}$.

Also, each of the images formed by the lens could act as an object for the reflecting surface of the lens which acts as a convex mirror.

So that would mean an infinite string of images (in theory), of lessening brightness, yes, but images nonetheless.

My question

Does this unending formation of images actually take place?

Or am I wrong in my assumptions somewhere?

If I am, I'd like to stand corrected.