Police witness a twenty-five year old man fleeing from a bank robbery scene. No one was injured during the robbery.

Fact pattern: Police witness a twenty-five year old man fleeing from a bank robbery scene. No one was injured during the robbery. The man is carrying a gun as he runs through the streets. They follow him on foot to a home several blocks away from the bank that the officers know regularly house several young children. They immediately follow him into the house and arrest him in the foyer of the house. He is charged for the bank robbery and found guilty by a jury. Although a robbery conviction usually carries a sentence of 15 years in this state, the judge sentenced the man to life in prison with no chance of parole. Please discuss (i) whether the arrest of the man in a private home without a warrant violates the Fourth Amendment and (ii) whether the life sentence violates the Eighth Amendment. Fully explain your answers from a constitutional perspective.