This is an exercise to show that completeness is essential for the fixed point property of a contraction mapping
Consider $X=\{P(x)=\sum_{n=1}^N a_nx^n,a_n\in \mathbb{R}\}$ the real-coefficient polynomials of degree less than $N$ and the norm $||P||=max\{|a_n|\}$.
The problem claims that the mapping $$T:p(x)\longrightarrow (1+x)p(x)$$ is a contraction but i can't see why since for $$p(x)=x^2+x,q(x)=x^2$$ we have $$||T(p-q)||=||x^2+x||=||p-q||=||x||$$.
any hints would be great.