what was king George and the British military trying to prove to the Americans when they attempted to arrest Adams and Hancock

The attempt to arrest Adams and Hancock was to show the Americans that whoever dares to betray or commit treason against the king and the government is a traitor and must be arrested and punished.

Explanation:

John Adams and Samuel Hancock were two patriots or rebel leaders working to dismantle and push forward for the independence of America from the British. And so, it was important for the British government to ensure that they don't spread their influence to more people.

In their act of propagating independence ideas, John Adams and Samuel Hancock were treated as traitors, working against the monarchy and the government. They were labeled betrayers of King George and thus, were ordered to be arrested. In doing do, the king as well as the British military is making their assertion that whoever betrays or works against the British authority's interest is a traitor for the whole nation and thus, must be arrested and kept away from spreading more "lies" against the King and authority.

usually her father was the one that controlled it after the death of the husband