Cauchy sequences and infinite series

Question

I'm working through Spivak's Calculus. In the chapter for Infinite Series, Spivak says that a sequence a_n is summable if and only if lim_{m,n —> ∞} s_m – s_n = 0 (where s_m = a₁ + a₂ + ... + a_m). Alright, so far so good.

But then Spivak says that this is equivalent to the "Cauchy Criterion":

Sequence a_n is summable iff lim_{m,n —> ∞} a_n+1 + ... + a_m = 0.

And he says that this entails the "Vanishing Condition":

If {a_n} is summable, then lim_{n —> ∞} a_n = 0.

What I don't understand is why the first statement entails the Cauchy Criterion and the Vanishing Condition.

Thank you.

Answer 1

A sequence $\{x_n\}_{n=1}^{\infty}$ is called Cauchy if, for every $\epsilon>0$, there exists $N\in\Bbb N$, such that $\forall m,n>N$, we have $|x_m-x_n|<\epsilon$.

We have that a real sequence converges if and only if it is Cauchy.

Now for a series $\sum_{n=1}^{\infty}a_n$. The series is defined if the sequence of partial sums $\{s_n\}_{n=1}^{\infty}$, defined by $s_n=\sum_{i=1}^n a_i$, converges, or equivalently if $\{s_n\}_{n=1}^{\infty}$ is Cauchy.

Now for $m>n$, $$|s_m-s_n|=s_n=\left|\sum_{i=1}^m a_i-\sum_{i=1}^n a_i\right|=\left|\sum_{i=n+1}^m a_i\right|.$$ Also we have $$|s_{n}-s_{n-1}|=|a_n|.$$ Hence $\{s_n\}_{n=1}^{\infty}$ is Cauchy, means for every $\epsilon>0$, there exists $N\in\Bbb N$, such that for all $m>n>N$, $|s_m-s_n|=\left|\sum_{i=n+1}^m a_i\right|<\epsilon$.

And this implies for $n>N+1$, $|a_n|=|s_n-s_{n-1}|<\epsilon$.