Peter said:
Thanks steep ...
... here is an attempt (which I fear is a bit rough around the edges ...)To show $$\sum x_n$$ is convergent using the Cauchy Criterion (for the sequence of partial sums ...) we need to show ... ...... that for each $$\epsilon \gt 0 \ \exists$$ an integer N such that $$\ \forall \ m, n \geq N$$ we have $$\mid s_m - s_n \mid \lt \epsilon$$ ... ...Assume $$m \gt n$$ ... then ...$$\mid s_m - s_n \mid \ = \ \left\vert \ \sum_{ k= n + 1 }^m x_k \ \right\vert $$ $$\Longrightarrow \mid s_m - s_n \mid \ \leq \ \sum_{ k= n + 1 }^m \mid x_k \mid$$ ( I think this step is correct... ... maybe this was the time to use Triangle Inequality ? )$$\Longrightarrow \mid s_m - s_n \mid \ \leq \ \sum_{ k= n + 1 }^m c_n \text{ for } n \geq N_0$$ $$\Longrightarrow \mid s_m - s_n \mid \ \lt \ \epsilon$$ since $$\sum c_n$$ is convergent ( and hence $\displaystyle\lim_{ n \to \infty} c_n = 0 )$
I am very dubious that above is correct as I didn't use the triangle inequality ... which is supposedly necessary ...
Peter
You
did use triangle inequality, on the second line where you wrote "I think this step is correct..." that is precisely the triangle inequality.
if it helps, you can re-write it in vector algebra as
$\left\vert \ \sum_{ k= n + 1 }^m x_k \ \right\vert$
$ = \big \Vert \mathbf x_{n+1} + \mathbf x_{n+2} + ... + \mathbf x_m \big \Vert_1$
$ \leq \big \Vert \mathbf x_{n+1}\big \Vert_1 + \big \Vert \mathbf x_{n+2} \big \Vert_1 + ... + \big \Vert \mathbf x_m \big \Vert_1 $
$= \sum_{ k= n + 1 }^m \mid x_k \mid $
where $\mathbf x_k = x_k \cdot \mathbf e_1$ (the 1st standard basis vector in say $\mathbb C^2$). Writing it out like this isn't needed but, may make it more obvious that you are applying triangle inequality. = = = =
so taking what you have, I suppose I'd write it as $$\mid s_m - s_n \mid \ $$
$$ = \ \left\vert \ \sum_{ k= n + 1 }^m x_k \ \right\vert $$
$$ \leq \ \sum_{ k= n + 1 }^m \mid x_k \mid$$
$$= \sum_{ k= n + 1 }^m c_n $$
$$\lt \ \epsilon$$
i.e. we know the partial sums for $c_n$ are Cauchy Sequences because the we know the series involving $c_n$ converges to a finite limit (and one implies the other in single variable analysis).
= = = =
edit:
one word of caution -- I didn't follow why you said
$$\sum c_n$$ is convergent ( and hence $\displaystyle\lim_{ n \to \infty} c_n = 0 )$
and in particular that $c_n \to 0$
The point really is that $c_n \to 0$ rapidly -- i.e. the series converges and hence the sequences of partial sums involving $c_n$ are Cauchy.