Let V consist of all sequences [x0, x1, x2,...) of numbers, and define vector operations [x0. x1,.....)

Question:

Let V consist of all sequences [x0, x1, x2,...) of numbers, and define vector operations
[x0. x1,.....) + [y0, y1,.....) = [x0 + y0, x1 + y1,.....)
r[x0, x1...) = [rx0 > rx1...)
Define T: V → U and S: V → V by T[x0, x1,...) = [x1, x2,...) and S[x0, x1, ......) = [0, x0, x1,...). Show that TS = 1v, so TS is one-to-one and onto, but that T is not one-to-one and S is not onto.