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Theorems 2 and 3: ε, δ Condition for Limits
Like other types of limits, limits of sequences have an ε, δ condition. It will be used later to prove theorems on series. THEOREM 2 (ε, δ Condition for Limits of Sequences) lim_{n→∞} a_{n} = L if and only if for every real number ε > 0 there is a positive integer N such that the numbers a_{N}, a_{N+1}, a_{N+2}, ..., a_{N+m},... are all within ε of L. The proof is similar to that of the ε, δ condition for limits of functions. The ε, N condition says intuitively that a_{n} gets close to L as the integer n gets large. A similar condition can be formulated for lim_{n→∞} a_{n} = ∞. THEOREM 3 (ε, δ Condition for Infinite Limits) lim_{n→∞} a_{n} = x if and only if for every real number B, there is a positive integer N such that the numbers a_{N}, a_{N+1}, a_{N+2}, ..., a_{N+m}, ... are all greater than B. We conclude this section with another useful criterion for convergence. CAUCHY CONVERGENCE TEST FOR SEQUENCES A sequence <a_{n}> converges if and only if (1) a_{H} ≈ a_{K} for all infinite H and K. PROOF First suppose <a_{n}> converges, say lim_{n→∞} a_{n} = L. Then for all infinite H and K, a_{H} ≈ L ≈ a_{K}. Now assume Equation 1 and let H be infinite. There are three cases to consider.


Home Infinite Series Sequences Theorems 2 and 3: ε, δ Condition for Limits 