Use the definition of convergence to prove the given limit....

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Key Concepts

Epsilon-Definition of Convergence

The epsilon-definition of convergence asserts that for every positive number ?, there exists a natural number N such that for all indices n greater than N, the absolute difference between the sequence term and the limit is less than ?. This rigorous formulation is frequently used to formally prove the convergence of sequences, as it precisely quantifies 'getting arbitrarily close.'

Sequence Convergence

Sequence convergence refers to the concept that as the sequence index increases, the sequence values get arbitrarily close to a fixed number, known as the limits. In this context, proving a sequence converges to a limit involves showing that the difference between the sequence terms and the limit can be made smaller than any positive real number provided the index is large enough.

Bounded Functions

The concept of bounded functions is crucial when dealing with functions like sine, which oscillate but remain within a fixed range. Specifically, the sine function is bounded between -1 and 1. This boundedness is often exploited in convergence proofs, as it allows one to apply inequalities and comparisons to establish that the sequence terms converge to a particular number.

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