I have been reading a paper written by J.Bourgain and I am stuck at one inequality.

So the author is trying to estimate the $L^2$ norm of

$$F(t)=sum_{min mathbb{Z}} e^{2imDelta t}(f_m overline{f}_{m+Delta})(t)$$

where $Delta in mathbb{Z}$ be given and $(f_m)$ is a sequence of functions of time $t$.

And he defined, for $jgeq 0$,

$$f_{m,j}(t)=sum_{|n| sim 2^j} widehat{f}_m(n)e^{int}hspace{1cm} ; f_m=sum_j f_{m,j}$$

using Littlewood-Paley decomposition.

**And he suddenly says that, by using triangular inequality,**

$$left| F right|_{L^2(T)} leq sum_{jgeq k} left| sum_{min mathbb{Z}} e^{2imDelta t}(f_{m,j} overline{f}_{m+Delta,k}) right|_{L^2(T)} $$

**Here, he only sum them for $jgeq k$ ignoring the part of $j<k$.** I cannot see what I am missing. I hope I can have some hint from this website. Thanks in advance.