Because, honestly, what proves your existence more than a book with traces of you in it? At that precise moment, when you write even a single letter on its pages, you are the only one in the entire world who owns a book like that. That book is yours to claim as one of a kind because you wrote in it.
So make that book yours - your own personal means of understanding the world and the people in it. Scribble in it. Write notes in its margins. Underline words that resonate. Don't fear defacing your precious literature. In the end, what is an empty canvas worth? A lived-in book serves more purpose than a brand new one that appears as though it has never been touched.
And after all that, you'll find yourself feeling impossible to read that book the same way again, what more part with it, because it has become an integrated part of you. No more reading that particular book for kicks, because it'll be like meeting an old friend again.
This is, I'm sure, is what it means to read as a writer who reads.