Unbounded multipliers can preserve uniform convergence ($g(x)=1/x,0<x\le 1/2$)

An unbounded function can multiply a series while preserving uniform convergence. On $E=(0,1/2]$, let $g(x)=1/x$. The geometric series givesThe equality follows because $x^N/(1-x)$ increases up to $x=1/2$. Both the first term and the sum are unbounded, while their difference after sufficiently many terms is uniformly small. Uniform convergence controls approximation errors; boundedness of each individual function is a separate property.