If |ω|=1 , then the set of points z=ω+1ω is satisfies

As |ω|=1⇒|ω|2=1⇒ωω¯=1⇒1ω=ω¯Thus, z=ω+1/ω=ω+ω¯=2Re⁡(ω)Now,  |Re⁡(z)|=|2Re⁡(ω)|=|2Re⁡(ω)|≤2|ω|=2