Here's how to break it down:
* Infinitive phrase: "to be persistent"
* Infinitive: "to be"
* Subject: The entire infinitive phrase "to be persistent" is the subject of the sentence.
* Verb: "is"
* Predicate: "a sign of stubbornness"
The sentence essentially means: "Being persistent is a sign of stubbornness." The infinitive phrase is acting as the noun that is being described by the rest of the sentence.