This problem is an "all" type problem, where the connection between the conclusion to be proved and the conditions is not obvious, and the clues for directly deducing the conclusion from the conditions are not clear. Therefore, we consider using the method of contradiction. Assume that $a$, $b$, $c$ are not all positive numbers. In this case, we need to discuss the situations where $a$, $b$, $c$ are not positive numbers one by one. However, given the characteristic of the conditions (the conditions of the proposition do not change when $a$, $b$, $c$ are interchanged arbitrarily), we only need to discuss one of the numbers (for example, $a$), and the other two numbers (for example, $b$, $c$) are similar to this situation.

When the conclusion of a proposition appears in the form of "at most", "at least", "unique", or in a negative form, it is appropriate to use the method of contradiction for proof. The key to the method of contradiction is to derive a contradiction under correct reasoning. The contradiction can be: ① Contradiction with known conditions, ② Contradiction with assumptions, ③ Contradiction with definitions, axioms, theorems, ④ Contradiction with facts, etc. The method of contradiction is often a powerful tool for solving some "difficult" problems and is a powerful weapon in mathematical proofs. Reasoning and proof are general ways of thinking in mathematics, and they are also the basic skills for learning and doing mathematics.

Therefore, the proof method suitable for this problem is $\boxed{\text{contradiction}}$.