I think it would be a good idea to notice that selfishness and altruism are, what logicians call, contraries, but not, contradictories. Contraries can both be false (although cannot both be true). Contradictories cannot both be false nor can they both be true. For instance, an animal cannot be both a dog and a cat, but an animal may be neither a dog nor a cat, but something else. So dog and cat are contraries, but not contradictories. On the other hand, an animal cannot be both a dog and not a dog. So dog and not-dog are contradictories, not contraries.
In the same way, a person cannot be (in the same way, and in the same respect) both an altruist and selfish. But a person can be neither an altruist nor selfish.
A person may simply act in his own self-interest[i] without taking anything away from another that the other is entitled to have. If, for instance, a mother leaves two pieces of cake for her two children to eat, and if one of them takes one piece and leaves the other for the other child for whom it is meant, then that one has acted in his own self-interest, but has not been selfish. If, on the other hand, he takes both pieces of cake, the one meant for him, and the one meant for the other child, then the mother would be right to call him selfish. But she would not be right to call him selfish if he took only the piece meant for him, and left the other for the other child.
The term “selfish” is a negative moral judgement. That is why the Ayn Rand title, “The Virtue of Selfishness” is quite idiotic. It is what is sometimes called an oxymoron. What Rand means is “the virtue of self-interest.” But then, that would not have been as jazzy, and it would not have sold as many books.
But the important thing to recognize is that acting self-interestedly need not be acting selfishly. (And, of course, acting self-interestedly is not acting altruistically either.) When I am tired, and go to bed, I am acting in my self-interest, but unless another person is involved, and I am depriving him of something he is entitled to have, I am not acting selfishly.
Most of the time, we are all acting self-interestedly. Perhaps part of the time, altruistically, and part of the time, unfortunately, selfishly. But is is clearly false that all actions done in one’s own self-interest are selfish actions.