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某些与自然数有关的命题不易直接用数学归纳法证明,但有时却可以用数学归纳法证明比原命题更强的新命题.由于此强命题是使原命题成立的充分条件,因而就能达到间接证明原命题的目的。这种证法容易奏效的原因十分简单:较强的命题其归纳法假设也较强,所以有时会更方便。请看下面两例:
Some propositions related to natural numbers are not easily proved directly by mathematical induction, but sometimes it is possible to use mathematical induction to prove new propositions that are stronger than the original proposition. Because this strong proposition is a sufficient condition for the original proposition to be established, it can be achieved. Indirectly prove the purpose of the original proposition. The reason why this type of proof is apt to work is very simple: stronger propositions and their inductive assumptions are also stronger, so sometimes it is more convenient. Please see the following two examples: