long sighted person lens

A long-sighted person cannot see clearly on objects placed at the "least distance of distinct vision" (about 25 cm from the eye) because of the shortening of the eye ball. In other words, the person's 'near point'(which the nearest point from the eye that an object placed there can be seen clearly) is farther than 25 cm.
The purpose of wearing a pair of eye glasses (converging lens) is to correct the "near point", such that objects placed at the "least distance of distinct vision" form images at further distance from the eye ball. The ray diagram is just the one that shown by you on your question.
For objects placed further, the lens of the eye can be adjusted such that clear image is formed at the retina.
Therefore, the final image formed by a long-sighted person is not merely the image formed by the eye-glasses, but the image formed by the combination of the eye-glasses ( converging lens) and the lens of the eye (another converging lens). The focal length of the latter is adjustable to the change in distance of the object.
Hence, just simply considering the ray diagram of the eye-glass doesn't reflect the real situation.