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,
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
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