Example 1 The elements of set $A$ are all integers, the smallest of which is 1, and the largest is 100. Except for 1, each element is equal to the sum of two numbers (which can be the same) in set $A$. Find the minimum number of elements in set $A$.


Solution: Construct a set with as many elements as possible to satisfy the conditions, such as $\{1,2,3,5,10,20,25,50, 9\}$.

Extend $\left\{1,2, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, 100\right\}$ to also satisfy the conditions, then $x_{1} \leqslant 4, x_{2} \leqslant 8, x_{3} \leqslant 16, x_{4} \leqslant 32, x_{5}$ $\leqslant 64$.
$$
\text { but } \begin{array}{l} 
x_{4}+x_{5} \leqslant 96<100, \therefore x_{5}=50, \\
x_{3}+x_{4} \leqslant 48<50, \therefore x_{4}=25, \\
x_{2}+x_{3} \leqslant 24<25, \therefore 25=2 x_{3} .
\end{array}
$$

Contradiction.
Therefore, the minimum number of elements in $A$ is 9.