Example 8 If $a, b, c$ are the length, width, and height of a rectangular prism, and $a+b-$ $c=1$, it is known that the length of the diagonal of the rectangular prism is 1, and $a>b$, try to find the range of values for the height $c$.

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Analysis: From the formula for the length of the diagonal of a cuboid, we get $a^{2}+b^{2}+c^{2}=1$

Thus $\left\{\begin{array}{l}a^{2}+b^{2}=1-c^{2}, \\ a+b=1+c .\end{array}\right.$
$\because\left(\frac{a+b}{2}\right)^{2}b)$,
$\therefore\left(\frac{1+c}{2}\right)^{2}0)$,
Solving, we get $0<c<\frac{1}{3}$.
The transformations of the mean inequality are diverse, and using them flexibly makes problem-solving concise and clear.