Part 4/8:
The number line representation can be illustrated with various ( \epsilon ) values shrinking over time. Each time ( \epsilon ) is reduced, a new integer ( N ) is found that accommodates this tighter binding, thereby demonstrating the converging nature of the sequence toward ( L ). The principle holds—no matter how small ( \epsilon ) becomes, there exists an ( N ) guaranteeing that all higher terms of the sequence fall within the specified limits.
This rigorous approach emphasizes the necessity of making arguments about limits tighter and more precise rather than using vague terms like "sufficiently large."