What is the proof in the play of the nobility of death?

Perhaps more than in any of his other works, in Julius Caesar, Shakespeare has his characters repeatedly and forcefully state their defiance of death. There is, partly, a historical reason for this. Though the anachronisms in Shakespeare's plays have been noted over and over again by carping critics, he portrays the ancients correctly, for the pagan religions didn't express the fear of punishment in the afterlife that came about in a later time, and that is expressed in Macbeth, Hamlet, and other plays taking place within the Christian era. As in much of Shakespeare, however, a central theme is the transient nature of earthly concerns. Caesar has no fear of going to the Senate though he's been warned that something will happen to him on the Ides of March:

Of all the wonders that I yet have heard,
It seems to me most strange that men should fear,
Seeing that death, a necessary end,
Will come when it will come.

The conspirators, though they despise Caesar, express the same point of view. First, death is noble to them because they believe their act of assassination will be memorialized by posterity, as Cassius says, "How many times will this our noble scene / Be played again in states unborn and accents yet unknown," and in the line "Why, he who cuts off twenty years of life / Cuts off so many years of fearing death," the sentiment is basically that this life is imperfect, wracked by irrational fear, and that death is somehow a condition to be preferred over earthly life.
In his soliloquy over Caesar's corpse, Antony does not appear to sentimentalize death. But in saying, "O pardon me thou bleeding piece of earth / That I am meek and gentle with these butchers," he states the view that in death we all return to the earth, which in Greco-Roman mythology is the mother of all. If Caesar were, as Antony calls him, "the noblest man that ever lived in the tide of times" death has, in fact, ennobled him further. To Antony, and to the Roman populace, Caesar has become a martyr, a symbol of a man who, in spite of being born into the upper class and having usurped the power of the Senate, had the interests of the common people at heart. Had Caesar not been murdered, it's an open question whether he would have attained this status.
At the end, the words spoken by Antony over Brutus are a kind of blessing, one which similarly confers noble and immortal status upon Brutus, though he was part of the conspiracy that brought about the catastrophe. Even Octavius, Caesar's grand-nephew, directs Brutus's body to be laid honorably in his tent. Had Brutus been still alive and captured, it is doubtful that he would have been treated so charitably. Cassius, though lacking the pure motives of Brutus, also, in his own death scene, becomes a more sympathetic character to the audience than he had been up to that point, when he speaks gently to his servant and recollects, "In Parthia did I take thee prisoner." The characters who die become ennobled by their passing and by their courage in having faced death without regrets.