Completeness of Vector Space

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