The augmented simplicial chain complex of a simplex is contractible

Define $h_{-1}:\mathbb{Z}\to C_0({\mathrm{\Delta }}^n)$ by $h_{-1}(1)=[a]$. For an oriented simplex $[v_0,\dots ,v_k]$, set $h_k[v_0,\dots ,v_k]=[a,v_0,\dots ,v_k]$ if $a\mathrm{\notin }\{v_0,\dots ,v_k\}$ and set $h_k[v_0,\dots ,v_k]=0$ if $a\in \{v_0,\dots ,v_k\}$. Because adjoining $a$ to a face of a simplex still gives a face of ${\mathrm{\Delta }}^n$, each $h_k$ is well defined.

If $a\mathrm{\notin }\{v_0,\dots ,v_k\}$, then the extra face created by applying $\mathrm{\partial }{h}_k$ and deleting $a$ is exactly $[v_0,\dots ,v_k]$, while every other face cancels with the corresponding term in $h_{k-1}\mathrm{\partial }$. If $a$ is already among the vertices, then $h_k$ is $0$ and the same cancellation leaves the identity term. Thus $\mathrm{\partial }h+h\mathrm{\partial }=\mathrm{id}$ on the augmented complex.

The family $(h_k{)}_{k\ge -1}$ is therefore a contracting homotopy, so the augmented simplicial chain complex of ${\mathrm{\Delta }}^n$ is contractible.