Gbeatty is quite right, in my opinion! Here are two quotes that you've request to support our contention:
In 1.1.86-90, Hamlet is lamenting the fate of his dear and good father, the former king. He is also cursing his uncle, Claudius, who has wrongly assumed the throne by murdering King Hamlet, and the betrayal of his mother, Gertrude. He cries out:
O horrible, O horrible, most horrible!
If thou hast nature in thee, bear it not.
Let not the royal bed of Denmak be
A couch for luxury and damned incest.
In the second line, he is challenging himself to act. The last, referring to incest, is in regard to his mother sleeping with her husband's brother. Tragic for the young man, indeed.
Another example of supreme tragedy is the fate of Ophelia, who has lost her father at the hand of her lover, Hamlet. In 4.7.210-212, her body is discovered by Gertrude. Laretes reflects on the tragedy, saying: Too much of water hast thou, poor Ophelia." He means that life became too much for her. The water literally took her life, but the psychological pressure is really what did her in.
Would you consider Shakespeare's, Hamlet a tragic hero? Why or why not?
Yes, I would consider Shakespeare's Hamlet a tragic hero. This is because a tragic hero is traditionally a person of good character or reputation and high position who is put in an untenable position where he fails due to a specific flaw or weakness in that character. Hamlet is good, and of high repute (a prince), but fails due to the uncertainty in his character.
"tragic hero is a character in a work of fiction (often the protagonist) who exhibits a tragic flaw, also known as fatal flaw, which eventually leads to his or her demise or defeat. The concept of the tragic hero was created in ancient Greek tragedy and defined by Aristotle (and others). Usually, this includes the realization of fatal flaw, which results in catharsis or epiphany. The tragic flaw is sometimes referred to as an Achilles' heel after the single fatal flaw of the Greek warrior Achilles. " (from http://en.wikipedia.org/wiki/Tragic_hero)
"Definition of a Tragic Hero
A tragic hero has the potential for greatness but is doomed to fail. He is trapped in a situation where he cannot win. He makes some sort of tragic flaw, and this causes his fall from greatness. Even though he is a fallen hero, he still wins a moral victory, and his spirit lives on." (from http://www.teachtheteachers.org/projects/JZarro2/process2.html)