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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)

limn→∞ an = L

if and only if for every real number ε > 0 there is a positive integer N such that the numbers

aN, aN+1, aN+2, ..., aN+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 an gets close to L as the integer n gets large. A similar condition can be formulated for limn→∞ an = ∞.

THEOREM 3 (ε, δ Condition for Infinite Limits)

limn→∞ an = x

if and only if for every real number B, there is a positive integer N such that the numbers

aN, aN+1, aN+2, ..., aN+m, ...

are all greater than B.

We conclude this section with another useful criterion for convergence.


A sequence <an> converges if and only if

(1) aH ≈ aK for all infinite H and K.


First suppose <an> converges, say limn→∞ an = L. Then for all infinite H and K,

aH ≈ L ≈ aK.

Now assume Equation 1 and let H be infinite. There are three cases to consider.

Case 1

aH is finite. Then for all infinite K, st(aK) = st(aH), so the sequence converges to st(aH).

Case 2

aH is positive infinite. For each finite m, aH ≥ am + 1. Among the hyper-integers {1, 2, ...,H-1}, there must be a largest element M such that aH ≥ aM + 1. But this largest M cannot be finite, and since aM not amost equal to aH, M cannot be infinite. Therefore Case 2 cannot arise.

Case 3

aH is negative infinite. By a similar argument this case cannot arise. Therefore, only Case 1 is possible, whence <an> converges.

Last Update: 2006-11-07