Is Beowulf Selfish?

I do believe Beowulf is selfish in his constant quest for fame. It seems that all he is ever concerned about is becoming famous and accomplishing feats simply to prove he is better than everyone else. He never says he wants to defeat Grendel or help Hrothgar because it's the right thing to do. It seems that to Beowulf, there is no good or bad, only strong or weak. The poet writes, "But Beowulf longed only for fame, leaped back into battle" (1529-1531) Beowulf does not leap back into battle because he longed to conquer evil, or because he knew it was the right thing to do. He leaped into battle once again so that his name would be remembered by posterity.
In my opinion, people should gain fame because they want to do the right thing, not do the right thing because they want to gain fame.