It is not enough to just sort ALL Contacts alphabetically to try to find a Contact whose Name starts with some letter.

Each letter on the A-Z List on the right side of the UI should be a link to the Contacts whose First Names begin with that letter.

Alternatively, along the top of the screen, create an A-Z horizontal List where each letter, when clicked, brings the User to the beginning of that list of names that start with that letter AND make the numeric list (at the bottom of the screen) limited to those pages of the list that begin with that letter.