Explain how the two prophesies that Macbeth thought never would come true actually do come true in The Tragedy of Macbeth, Act V.

There are four prophecies involved in act V. Three of them Macbeth thinks that they are impossible, but only because the witches convinced him to believe what he wanted.

In the first apparition, the witches warn Macbeth about Macduff.

"Macbeth! Macbeth! Macbeth! Beware Macduff.

Beware the thane of Fife. Dismiss me. Enough." Act IV Scene 1 Line 73

In the second apparition, the witches tell Macbeth that he shouldn't fear men, because no one that has ever been born from a woman will ever be able to kill him.

"Laugh to scorn the power of man, for none of woman born shall harm Macbeth" Act IV Scene 1 Line 81

This prophecy becomes true when Macduff tells him that he was not born naturally, that means that he was not actually born from a woman, so he is the only one that can and will kill Macbeth.

"Despair thy charm, And let the angel whom thou still hast served

Tell thee, Macduff was from his mother's womb Untimely ripped" Act V Scene 8 Line 13

This apparition along with the first one are related to the prophecy of Macduff killing Macbeth. Macduff was not born naturally, his mother had a cesarean (c-section), therefore this means that he was not born from a woman.

In the third apparition, there's a bloody baby wearing a crown and holding a branch from a tree. The witches tell Macbeth that he shouldn't worry about the battle because no one will defeat him until Birnam Woods starts moving towards his castle.

"Macbeth shall never vanquished be until Great Birnam Wood to high Dunsinane Hill shall come against him". Act IV Scene 1 Line 96

This prophecy becomes true when the soldiers, that were gathering in Birnam Wood, decide to grab braches of trees to hide themselves from Macbeth and his soldiers, when Malcom (the rightful king of Scotland, son of Duncan - the king killed by Macbeth) gives the order, all the soldiers holding their branches have to start running towards Macbeth's castle (Also known as Dunsinane); this is seen by Macbeth from the castle, a messenger comes and tells him that he saw the Birnam Wood moving, that's when Macbeth starts feeling afraid for the first time.

(A messenger to Macbeth) "As I did stand my watch upon the hill, I looked toward Birnam, and anon methought the wood began to move." Act V Scene V Line 33

When Macbeth is given these news, he realizes that the time to fight has come, and that the witches have tricked him into believing he was invincible, but now he is losing his confidence.

"I pull in resolution and begin to doubt th' equivocation of the fiend

That lies like truth. "Fear not, till Birnam wood

Do come to Dunsinane"; and now a wood comes toward Dunsinane.-Arm, arm, and out!-" Act V Scene V Line 41 This means that he is beginning to loose the confidence he had before, and he is doubting the words of the witches.

In the fourth apparition, Macbeth sees a succesion of kings, all of them looking similar to Banquo, this is a reference to the prophecy given to Banquo at the beginning of the play.

Macbeth is indeed killed by someone "not of woman born"-apparently, Macudff was delivered in a Caesarian section and so, in an era when this medical procedure was uncommon, seems not to be "born" in the usual biological way. Macbeth is also killed when Birnam wood comes to Dunsinane, since the soldiers attacking Dunsinane camouflage themselves with tree boughs that make it look like Birnam wood is approaching.