Example 7 When $x$ is a real number, we have $0 \leqslant\{x\}<1$.

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Let $x=n+y$, where $n$ is an integer and $0 \leqslant y<1$. From (II) and (III) we have
$$[x]=[n+y]=n+[y]=n \text {. }$$

From property (I) and equation (33) we have
$$\{x\}=x-[x]=n+y-n=y .$$

Since $0 \leqslant y<1$ and equation (34), we get $0 \leqslant\{x\}<1$.