This is a follow up question to Farcher's answer for the question - How does a Galilean telescope form an enlarged image even though it has a diverging lens?.
Let us consider the following ray diagram which shows a simple model of a Galilean telescope:
Image Source : Concepts of Physics by Dr. H.C.Verma, chapter "Optical Instruments", page 424, topic "Telescopes", sub topic "Galilean Telescope"
The following statement is from the book mentioned above:
If the telescope is set for normal adjustment, the final image $P''Q''$ is formed at infinity. Then $P'E=-f_e$ [where $f_e$ is the focal length of the eye piece] […]
$P'Q'$ is the image formed by the converging lens $L$. $P'Q'$ acts as an object for the diverging lens (eye piece). And it's said that for normal adjustment $P'Q'$ is at the focus of the bi-concave lens and the image $P''Q''$ forms at infinity.
In other words, the diverging lens forms an image at infinity for an object placed at its focal point. Isn't this a behaviour of a converging (convex) lens? This fact troubled me a lot, and I constructed the following ray diagram:
I've neglected the convex lens for the sake of simplicity.
It can be seen that the image $A'B'$ is formed at the midpoint of focal length on the same side of object $AB$ (image formed by the convex lens). I also verified it using the thin lens formula $\frac 1 v -\frac 1 u=\frac 1 f$. So for an object at the focal point of a diverging lens, the image forms midway between the object and the lens. But this is contradictory to what is being explained in my textbook, and in the answer linked above regarding Galilean telescopes.
In short, my question is - How does a diverging lens in a Galilean telescope form an image at infinity when its object is at its focal plane?