Washington and Rochambeau had General Cornwallis and his troops surrounded and greatly outnumbered at Yorktown, VA.
Oh no! We're outnumbered!
After about 11 days of battling, Cornwallis surrendered to the Patriots.
On October 19, 1781 Cornwallis signed the British surrender document which was formally known as the Articles of Capitulation.
On September 3, 1783, the Treaty of Paris was signed. This recognized the United States as an independent nation- from then on the Americans lived independently and were finally able to begin working on their brand new nation.