When in trouble with the Cyclops Odysseus made the Cyclops drunk so he could kill him easier and that would make it possible for him and his men to escape
Odysseus knew the trouble the sirens would cause, so he plugged his men ears with beeswax, but tied himself down so he could know when his men were safe from the sirens. So Odysseus proves to be hero because he is willing to suffer so that his men will be okay.
Odysseus dresses up as a old man and beats all of the other men to prove to his wife that he is who says he is.. but she still has some doubt.
You can't move our bed stupid
Odysseus kills all of the men that tried to take his wife knowing that they were married, then he made the maids clean up the mess and then killed them too.
Penelope realizes that Odysseus is her husband when she says she is going to move the bed, but Odysseus knows she cant move it because they made it in the tree and built the house around it. I think this most proves he's a hero because hero's always accomplish their missions no matter what it takes.