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.
In particular, we have :
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
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??
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.
And in the final steps, you juste have to keep instead of
Again, excuse me for my mistake