Question

Convergence and divergence of sequences?

The sequence converges to the number L if to every positive number epsilon there corresponds an integer N such that for all n, n>n |an-L|<epsilon.
Are the terms in the interval (L-ε, L+ε) the infinite elements of the sequence which are approaching the limit, L. Any elements to the right or left of that interval may not be converging to the limit but if we are studying the limit of a sequence as the number of terms increase to infinity the behavior of those finite points should not be used to decide on whether the sequence converges or diverges because we are taking a finite sample of that set. So my choice of epsilon depends on when the terms start to converge? Why is this definition any helpful? There an Infinite number of points within our choice of the closed interval and we are studying what limit L these points converge to?

Answer 1

See below.

Explanation:

Among other things, the definition says that for any positive `epsilon`, there are only finitely many points of the sequence (at most, `a_1` up to `a_N`) outside the interval `(L-epsilon,L+epsilon)`.

There are not infinitely many points outside of `(L-epsilon,L+epsilon)` (either left and right or above and below, depending on you image).

The person proving that the limit is `L` does not choose the epsilon. The opponent chooses the `epsilon`. The prover's job is to show that no matter which epsilon he is given, there is an `N` s.t. `n > N` implies `abs(a_n - L) < epsilon`