The fragment "without hope of succeeding" starts with the preposition "without". This is a prepositional phrase. In order to turn this fragment into a complete sentence, it needs both subject and predicate since they are both absent in the prepositional phrase. You can then add them and say, for example: With hope of succeeding, the man ran as fast as he can towards the finish line. In this sentence, we have the subject "man" and the predicate " ran as fast as he can". The sentence is complete.
