For a finite simplicial complex $X$, let $b_{i}(X)$ denote the rank of the finitely generated abelian group $H_{i} X$. Define the Euler characteristic $\chi(X)$ by the formula

$$\chi(X)=\sum_{i}(-1)^{i} b_{i}(X) .$$

Let $a_{i}$ denote the number of $i$-simplices in $X$, for each $i \geqslant 0$. Show that

$$\chi(X)=\sum_{i}(-1)^{i} a_{i}$$