Inequality regarding Littlewood-Paley decomposition – Mathematics Stack Exchange

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. I cannot see what I am missing. I hope I can have some hint from this website. Thanks in advance.