This is so straightforward and simple an assertion of the rationality of diachronic identity that it requires no gloss and can be reproduced directly. Incidently Sidney Shoemaker has a similar argument in his book ‘Self-Knowledge and Self-Identity’.
Who denies diachronic identity anyway? David Hume asks himself the question: How is a series of conscious states conscious of itself as a series? His bundle theory of states of consciousness creates this problem which it is unable to solve.
It is important to note that Sankara is not offering memory as a criterion of personal identity. He is merely using memory as sufficient to refute the momentariness (annica) doctrine of Buddhism namely that there is no connection between the string of conscious states that are had by a person. His self identity is pure consciousness itself that remains unchanging through all states of awareness including that of deep dreamless sleep known as ‘sushupti’ in Vedanta. Follow the topic heading on Deep Sleep for more notes on that angle unique to Advaitic Vedanta.
Sankara’s Commentary on the Brahma Sutras in which he refutes Buddhist Realism (B.S.B. II.ii.25)
25. And (a permanent soul has to be admitted) because of the fact of remembrance (i.e. memory).
Moreover, when the nihilist asserts all things to be momentary, he willhave to assert the perceiver also to be momentary. But that is an absurdity because of the fact of remembrance. Remembrance means recalling to mind something after its perception, and that can happen only when the agent of perception and memory is the same; for one person is not seen to remember something perceived by another. How can there be an awareness of the form, "I who saw earlier see now", arise unless the earlier and later perceiver be the same? Moreover, it is well known to all that direct experience in the form of recognition, such as "I who saw that, see this now", occurs only when the agent of seeing and remembering is the same.