Let $(X, \rho)$ be a Complete metric space, such that $\sup\limits_{x,x'\in X}\rho(x,x') < \infty$.
Let $G=G(X)$ be the group of homeomorphisms of $X$ with itself and let $\hat{\rho}(f,g)=\sup\limits_{x\in X}\left(\rho(f(x), g(x)) + \rho(f^{-1}(x), g^{-1}(x))\right)$ be a metric on $G$.
Then is it true that $(G, \hat{\rho})$ is a Complete metric space?
I think you can argue as follows:
Step 1: Let $C(X)$ denote the space of continuous functions $X \rightarrow X$, and $d(f,g)$ the supremum norm on $C(X)$: $d(f,g) = \text{sup}_{x \in X}\rho(f(x),g(x))$. Then (you can prove that) $C(X)$ is a complete metric space with respect to this norm.
Step 2: Let $\{f_n\}_{n \in \mathbb{N}}$ denote a Cauchy sequence in $G(X)$. Since $\hat{\rho}(a,b) \geq d(a,b)$ for all $a,b \in G(X)$, the $f_n$ are also a Cauchy sequence in $C(X)$ with respect to the metric $d$, and so converge to some limit, which we'll call $f \in C(X)$. Similarly, since $\hat{\rho}(a,b) \geq d(a^{-1},b^{-1})$ for all $a,b \in G(X)$, the $f_n^{-1}$ are a Cauchy sequence in $C(X)$, and so converge to some limit, which we'll call $g \in C(X)$.
Step 3: To conclude, you need to show that $f$ and $g$ are inverses, which means that $f \in G(X)$, and also that $f_n \rightarrow f$ in the $\hat{\rho}-$norm.
Let me know if you want help with the details.