How to reduce the compact support to the case of small diameters in Tao's "A sharp bilinear restriction estimate for paraboloids"

Question

I am reading Terence Tao's paper "A sharp bilinear restriction estimate for paraboloids" to prove the bilinear restriction estimate on paraboloids. In Section 3, he assumes that $\text{diam}(S_1),\text{diam}(S_2)\ll \text{dist}(S_1,S_2)$. My question is: how do we reduce the case of large $S_1,S_2$ with narrow but positive distance to the assumed case?

My approach is to decompose $S_1,S_2$ to coverings of almost disjoint compact subsets, whose diameters are much smaller than $\text{dist}(S_1,S_2), but got stuck.

Answer 1

Stupid me!

I posted my own answer anyway for other fledglings like me or for anyone to double check my proof!

($f_{1,i}$ is $f_1$ restricted to a small compact subset indexed as $i$. Similarly to $f_{2,j}$.