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## Main Question or Discussion Point

Theorem 3.23 is a very simple one: it says that if a series converges then the limit of the terms of the sequence is zero. However Rudin's way of justifying this fact doesn't seem valid to me. He uses the following logic:

A series converges if and only if the sequence of partial sums is cauchy meaning that for all ε > 0 there is an integer N s.t. for all n,m > N and n <= m the sum of the terms of the sequence from a_n to a_m is less than ε.

Rudin says that the case where n = m proves this theorem. However when n = m the only thing the cauchy criterion states is that the distance from a_n to a_n approaches zero. It does not actually say that the

To prove this we need the case where n = m - 1.

Then the difference between the two partial sums is a_m and therefore a_m approaches zero.

Am I wrong?

A series converges if and only if the sequence of partial sums is cauchy meaning that for all ε > 0 there is an integer N s.t. for all n,m > N and n <= m the sum of the terms of the sequence from a_n to a_m is less than ε.

Rudin says that the case where n = m proves this theorem. However when n = m the only thing the cauchy criterion states is that the distance from a_n to a_n approaches zero. It does not actually say that the

**value**of a_n approaches zero.To prove this we need the case where n = m - 1.

Then the difference between the two partial sums is a_m and therefore a_m approaches zero.

Am I wrong?