Example 12. For $>0, \frac{1}{b}-\frac{1}{a}=1$. Compare the sizes of the following four numbers: $\sqrt{1+a}, \frac{1}{1-\frac{b}{2}}, 1+\frac{a}{2}$,
$$
\frac{1}{\sqrt{1-b}}
$$
For these four numbers, if we use the method of finding differences to compare their sizes, we need to perform $C_{4}^{2}=6$ comparisons to get the answer. Can we first estimate the size relationship of these four numbers? We can use the method of substitution to test it.
$\because a>0$, let's assume $a=1$, from $\frac{1}{b}-\frac{1}{a}=1$, we get $b=\frac{1}{2}$.
Thus, $\sqrt{1+a}=\sqrt{2}, \frac{1}{1-\frac{b}{2}}=\frac{4}{3}$,
$$
1+\frac{a}{2}=\frac{3}{2}, \frac{1}{\sqrt{1-b}}=\sqrt{2} \text {. }
$$

From this, we can conjecture that for all $a, b$ values satisfying $a>0$,
$$
\begin{aligned}
\frac{1}{b}-\frac{1}{a}=1 \text {, we have: } \\
\frac{1}{1-\frac{b}{2}}<\frac{1}{\sqrt{1-b}}=\sqrt{1+a}<1+\frac{a}{2} .
\end{aligned}
$$