Thread: 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.

Hello,

Originally Posted by frater_cp

If (x_j) is a sequence in an inner product space X, such that the series
converges.

Show that is a Cauchy sequence , where

Let be the limit of

Hence,
In particular, we have :


So
Which is equivalent to

Since is a norm, it satisfies the triangle inequality :



But (assuming n>m) , where a is the norm of the unit vector.

Again, by the triangle inequality, we know that

So finally, we have (letting )



And this proves that is a Cauchy sequence

Hi Moo.

Thank you very much for your elegant answer.

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!

Hi

Originally Posted by frater_cp

Hi Moo.

Thank you very much for your elegant answer.

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...

...wrong

I'm sorry, I guess I was confusing myself... The norm of a number is its absolute value.
Here, is a number, a positive number, since a norm is always positive.

So

And in the final steps, you juste have to keep instead of

Again, excuse me for my mistake