# Thread: Sequence of partial sums is Cauchy

1. ## Sequence of partial sums is Cauchy

If (x_j) is a sequence in an inner product space X, such that the series
\|x_1\| + \|x_2\| + \cdots converges.

Show that (s_n) is a Cauchy sequence , where s_n = x_1 + \cdots + x_n.

2. Hello,
Originally Posted by frater_cp
If (x_j) is a sequence in an inner product space X, such that the series
$\|x_1\| + \|x_2\| + \cdots$ converges.

Show that $(s_n)$ is a Cauchy sequence , where $s_n = x_1 + \cdots + x_n.$
Let $S$ be the limit of $T_n=\sum_{j=1}^n \|x_j\|$

Hence, $\forall \epsilon>0,\exists N\in\mathbb{N},\forall k\geq N, \|T_k-S\|<\frac\epsilon 2$
In particular, we have :
$\begin{array}{ll} \forall m\geq N ~:~ \|T_m-S\|<\frac\epsilon 2 \\ \forall n\geq N ~:~ \|T_n-S\|<\frac\epsilon 2\end{array}$

So $\|T_n-S\|+\|T_m-S\|<\epsilon$
Which is equivalent to $\|T_n-S\|+\|S-T_m\|<\epsilon$

Since $\|.\|$ is a norm, it satisfies the triangle inequality :

$\|T_n-T_m\|=\|T_n-S+S-T_m\|\leq \|T_n-S\|+\|S-T_m\|<\epsilon$

But (assuming n>m) $\|T_n-T_m\|=\left\|\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|\right\|=a \left(\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|\right)$, where a is the norm of the unit vector.

Again, by the triangle inequality, we know that $\|s_n-s_m\|=\|x_n+x_{n-1}+\dots+x_{m+1}\|\leq \|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|$

So finally, we have (letting $\epsilon'=\epsilon/a$)

$\forall \epsilon'>0,\exists N\in\mathbb{N},\forall n,m\geq N, \|s_n-s_m\|<\epsilon'$

And this proves that $(s_n)$ is a Cauchy sequence

3. ## want to check something

Hi Moo.

I want to check something in your solution I do not understand.

Its were you state an equality (where assumed n>m) and then you have:
a = the norm of the unit vector

I do not understand this equality can you explain where this whole line comes from please??

Kind Regards!

4. Hi
Originally Posted by frater_cp
Hi Moo.

I want to check something in your solution I do not understand.

Its were you state an equality (where assumed n>m) and then you have:
a = the norm of the unit vector

I do not understand this equality can you explain where this whole line comes from please??

Kind Regards!
No problem. This proves that you're following what I've been doing...

$\left\|\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|\right\|=a \left(\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|\right)$
...wrong

I'm sorry, I guess I was confusing myself... The norm of a number is its absolute value.
Here, $\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|$ is a number, a positive number, since a norm is always positive.

So $\left\|\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|\right\|=\|x_n\|+\|x_{n-1}\|+\dots+\|x_{m+1}\|$

And in the final steps, you juste have to keep $\epsilon$ instead of $\epsilon'$

Again, excuse me for my mistake