Final answer:
The claim that both the Americans and the British gave up claims to the Mississippi for the sake of peace is false after the Treaty of Paris, Britain ceded these territories to the United States. The Treaty of Paris neglected the American Indian interests making this claim true.
Step-by-step explanation:
When examining the historical context of claims over the Mississippi River it is false that both the Americans and the British gave up claims to the Mississippi for the sake of peace. Following the American Revolutionary War and the signing of the Treaty of Paris in 1783 Britain recognized American sovereignty and relinquished its claims to the lands south of the Great Lakes and east of the Mississippi River.
The claim that the Treaty of Paris ignored the American Indians is also true. The treaty did not take into account the interests or land rights of the Indigenous peoples living in the territories discussed. Subsequently, conflicts arose between American settlers and Native American tribes due to this oversight.
Moving on to the expansion of the United States, it is true that the Louisiana Purchase doubled the territory of the United States. This acquisition in 1803 added a vast expanse of land to the United States, stretching from the Mississippi River to the Rocky Mountains and effectively doubling the size of the country.