In my experience writing formal letters, university applications, job applications etc, it is normal to start by stating the purpose of the letter using a sentence like:
"I am writing to apply for [job]"
"I am writing to present myself as a candidate for [university course]"
So your sentence is fine for the opening of a university application
Many schools require a "letter of motivation" or "Personal statement" attached to the application. Since it is expected and mandatory, stating the purpose is rather redundant. You can start off with something about yourself and the purpose or motivation for your application.