With f defined as $\displaystyle f(x)=e^x+6x-5$, show the fixed point iteration

$\displaystyle x_{n+1}=\frac{5-e^{x_{n}}}{6}$, $\displaystyle n=0,1,...$

must converge to $\displaystyle x_{*}$ with $\displaystyle x_{0} $\in$ [0,1]$

Had a go at this by using contraction mapping theorem and putting it in the form $\displaystyle f(x)=x-g(x)$, and as f(x)=0 then can work out what g(x) is but I'm not sure it is a contraction so not sure how can know it converges?