Example 4 Let $a$, $b$, $c$ all be real numbers no less than 3. Then the minimum value of $\sqrt{a-2}+\sqrt{b+1}+|1-\sqrt{c-1}|$ is $\qquad$
(2001, Hope Cup Junior High School Mathematics Competition Second Trial)
Given: $\because a \geqslant 3, b \geqslant 3, c \geqslant 3$,
$$
\therefore a-2 \geqslant 1, b+1 \geqslant 4, c-1 \geqslant 2 \text {. }
$$

Then the original expression $=\sqrt{a-2}+\sqrt{b+1}+\sqrt{c-1}-1$.
Also, $\because$ the smaller the radicand, the smaller its arithmetic root,
$\therefore$ the minimum value of the original expression is
$$
1+2+\sqrt{2}-1=2+\sqrt{2} \text {. }
$$