Final answer:
The claim that the Border States rapidly accepted Lincoln's gradual emancipation proposals and readily implemented the Thirteenth Amendment is false. The Border States resisted these changes due to their reliance on slavery.
Step-by-step explanation:
The statement that the Border States quickly accepted Lincoln's proposals for gradual compensated emancipation and willingly implemented the Thirteenth Amendment is false. Border States were initially resistant to Lincoln's proposals and did not quickly enact the Thirteenth Amendment. This amendment, which officially abolished slavery throughout the United States, was not eagerly adopted by these states, which had a significant interest in the institution of slavery. Additionally, the Emancipation Proclamation allowed Lincoln to take a critical step towards the elimination of slavery, but it did not fulfill any campaign promise to immediately eliminate slavery across the Union since it was limited in scope.