Example 3 Given that the line $l$ passing through point $P(1,2)$ intersects the positive half-axes of the $x$-axis and $y$-axis at points $A$ and $B$ respectively, find the minimum value of $|O A|+|O B|$.
Let the equation of line $l$ be $\frac{x}{a}+\frac{y}{b}=1(a>0$, $b>0)$, then $\frac{1}{a}+\frac{2}{b}=1$. By associating with the Cauchy-Schwarz inequality to construct the product of sums, we know that $|O A|+|O B|=a+b=(a+b)\left(\frac{1}{a}+\frac{2}{b}\right)$ $\geqslant(1+\sqrt{2})^{2}=3+2 \sqrt{2}$. Therefore, the minimum value sought is $3+2 \sqrt{2}$.