The first way Beowulf demonstrates courage is to take the initiative to sail across the sea with his fighting men to confront the monster Grendel. Although his family does owe Hrothgar a debt from former times, it would have been easy for Beowulf to have sidestepped this especially risky task as not his problem. Nevertheless, he sails off valiantly to meet the monster and defend civilization.
Beowulf also demonstrates his courage in the fearless way he meets and defeats Grendel, Grendel's mother, and the dragon. From youth to old age, he cannot be intimidated.
As Beowulf himself states, it is not talk of courage that matters. Instead, it is putting one's own body on the line to defend society. Beowulf says to the bragging Unferth:
if you were truly
as ... courageous as you claim to be
Grendel would never have got away with
such unchecked atrocity, attacks on your king,
havoc in Heorot and horrors everywhere.
What Beowulf means is that courage is the element that keeps barbarism at bay.
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