To prove that if $\varphi(x)$ satisfies equation (1), then we have
$$\begin{aligned} 4 \varphi\left(\frac{x_{1}+x_{2}+x_{3}+x_{4}}{4}\right) & \leqslant 2 \varphi\left(\frac{x_{1}+x_{2}}{2}\right)+2 \varphi\left(\frac{x_{3}+x_{4}}{2}\right) \\ & \leqslant \varphi\left(x_{1}\right)+\varphi\left(x_{2}\right)+\varphi\left(x_{3}\right)+\varphi\left(x_{4}\right) \end{aligned}$$

and so on. Thus, we have proved that for a special sequence of $n$, i.e., $n=2^{m}$, we have
$$\varphi\left(\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right) \leqslant \frac{\varphi\left(x_{1}\right)+\varphi\left(x_{2}\right)+\cdots+\varphi\left(x_{n}\right)}{n}$$

To prove that equation (4) holds universally, it suffices to show that if it holds for $n$, then it also holds for $n-1$. (Here we use reverse induction. For a more direct proof following Cauchy's method, refer to Jensen's related articles.) Thus, assume that equation (4) holds for $n$ numbers, and now consider $n-1$ numbers $x_{1}, x_{2}, \cdots, x_{n-1}$. Let $x_{n}$ be the arithmetic mean of these $n-1$ numbers (with equal weights), and apply equation (4), we get
$$\begin{aligned} \varphi(A) & =\varphi\left(\frac{(n-1) A+A}{n}\right)=\varphi\left(\frac{x_{1}+x_{2}+\cdots+x_{n-1}+A}{n}\right) \\ & \leqslant \frac{\varphi\left(x_{1}\right)+\varphi\left(x_{2}\right)+\cdots+\varphi\left(x_{n-1}\right)+\varphi(A)}{n} \end{aligned}$$

Thus, we have
$$\varphi(A) \leqslant \frac{\varphi\left(x_{1}\right)+\varphi\left(x_{2}\right)+\cdots+\varphi\left(x_{n-1}\right)}{n-1} .$$

Next, for non-negative rational numbers $r_{1}, r_{2}, \cdots, r_{n}, r_{1}+r_{2}+\cdots+r_{n}=1$, there exists a natural number $m$ and non-negative integers $p_{1}, p_{2}, \cdots, p_{n}$, such that $m=p_{1}+p_{2}+\cdots+p_{n}$, and $r_{i}=\frac{p_{i}}{m}$ $(i=1,2, \cdots, n)$. Now, according to equation (4), we have
$$\varphi\left(\frac{p_{1} x_{1}+p_{2} x_{2}+\cdots+p_{n} x_{n}}{m}\right) \leqslant \frac{p_{1} \varphi\left(x_{1}\right)+p_{2} \varphi\left(x_{2}\right)+\cdots+p_{n} \varphi\left(x_{n}\right)}{m},$$

Thus, $\varphi\left(\sum r_{i} x_{i}\right) \leqslant \sum r_{i} \varphi\left(x_{i}\right)$.
Finally, substituting the rational approximations $r_{i}$ for each $p_{i}$ and taking the limit, we obtain equation (3).