Proximal mapping and projection

Question

Show that for $x = [x_1; x_2; \ldots; x_n]$, $$ \max_{1 \le i \le n} \{x_i\} = \max_{y \in C} \{x*y\}, $$ where $C$ is the unit simplex defined by $$C = \{y \in \mathbb{R}^n| y > 0, 1^Ty = 1\}.$$ Hence, prove that the proximal mapping of $f : \Bbb R^n → \Bbb R$ defined by $f(x) = \max_{1≤i≤n}\{x_i\}$ is given by $$ P_f (x) = x − Π_C(x) $$ where $Π_C(x)$ is the projection of $x$ onto $C$