a) you have this written down wrong.

you want to show that θψ = 1_{R}but ψθ ≠ 1_{R}

b) what happens if you replace 0 in the definition of ψ with some other integer k?

c) how about π(a_{1},a_{2},a_{3},....) = (1,0,0,0,......) ?

d) suppose ρθ = 0, that is: ρθ(a_{1},a_{2},a_{3},....) = (0,0,0,....).

this means that ρ(a_{2},a_{3},....) = (0,0,0,......), no matter how we choose a_{2},a_{3},....

so ρ = 0.