Homeomorphism of the cone on a space The unreduced cone $\text{Cone}(X)$ on a space $X$ is given by $$ \text{Cone}(X)=X\times I/ X\times \{1\}$$ where $I$ is a unit interval.
Show that Cone($S^{n-1}$) is homeomorphic to $D^{n}$ where $S^{n-1}=\left\{x\in R^{n}| \left\| x \right\|=1 \right\}$ and $D^{n}=\left\{x\in R^{n}| \left\| x \right\| \leq 1 \right\}$.
Intuitively I want to construct a homeomorphism by projecting some part of the cone into the interior of the sphere. But I don't know how to carry it out by using accurate mathematical language.

Let $p$ denote the apex of the cone $\text{Cone}(S^{n-1})$. For each $x\in S^{n-1}$ and each $t\in[0,1]$ let $c(x,t)\in \text{Cone}(S^{n-1})$ satisfying 
$$ c(x,t)=(1-t)p+tx$$
Then let $f: \text{Cone}(S^{n-1})\to D^n$ be defined by
$$ f(c(x,t))=tx $$
This continuously maps $\text{Cone}(S^{n-1})$ onto $D^n$ and its inverse continuously maps $D^n$ onto $\text{Cone}(S^{n-1})$.
