Over 350,000 residents felt proud of their beautiful city.
Felt is the linking verb in this sentence. (Sometimes these verbs: appear, feel, grow, look, prove, remain, smell, sound, and taste are linking verbs; sometimes they are action verbs. If you can substitute am, is, or are for the verb and the sentence still sounds logical, you have a linking verb on your hands.)
Proud is the predicate adjective in this sentence because, again, it follows the linking verb, and modifies or directly refers to the subject of the sentence (residents).
