3.1. Sequences

Example 3.1.3(a):

The sequence converges to zero.

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= {1, 1/2, 1/3, 1/4, ... }, which seems to indicate that the terms are getting closer and closer to zero. According to the definition of convergence, we need to show that no matter which > 0 one chooses, the sequence will eventually become smaller than this number. To be precise: take any > 0. Then there exists a positive integer N such that 1 / N < . Therefore, for any j > N we have:

| 1/j - 0 | = | 1/j | < 1/N <
whenever j > N. But this is precisely the definition of the sequence {1/j} converging to zero.

While it looks like this proof is easy, it is a good indication for '-arguments' that will appear again and again. In most of those cases the proper choice of N will make it appear as if the proof works like magic.

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