Let

be the space of sequences

such that

, with inner product

and norm

.

I need to show that

is complete with respect to the inner product i.e. a Hilbert Space. I am assuming that

is complete.

I know that I need to show that every Cauchy sequence in H is convergent. I can see that this follows relatively easily from the definition of H.

So I have taken a sequence

but for this to be Cauchy

. But I can't see how to show that, given that the inner product on

has a

below it. Any help would be great. Thanks