Understanding the proof of theorem 6.4 of Rudin's Functional analysis

78 Views Asked by At

In Rudin's functional analysis text I do not understand the following step in theorem 6.4 (a).

$\phi -\phi _i \in (1- \delta _i ) W_i$

How do we get such delta? In particular how does $W_i$ being convex balanced set in $\mathcal D (\Omega)$ imply the existence of such $\delta$.
For reference I am attaching screenshot. Thanks in advance for the help. enter image description here