"It really mean nothing" seems to be a very accurate description of your attempt to make clear to others what appears to be a very vague idea in your...
When using induction to prove that the proposition is true for k + 1, you use the fact that the proposition is true for k (and perhaps also the fact...
EVERYONE makes mistakes. That is why people with much experience ALWAYS say that you SHOULD check your work. You may make mistakes, but you will not...