Why Does God Allow Sin To Remain?

Why does God save us and allow us to remain sinful? There are few "problems" more relevant and significant than this in the Christian experience. A person sins and is immediately confronted with the question, "How can I say I love God when I obviously don't love him enough to stop sinning?"

Some then wonder why the Lord allows sin to linger. Some wonder if he actually exists. Some wonder if their faith actually exists; or, whether they will soon be lugging around the label associated with hypocritic Christianity.