From my understanding of my topic, if a statement is true for $n = 1,$ and you assume a statement is true for arbitrary integer $k$ and show that the statement is also true for $k + 1,$ then you prove that the statement's true for all $n \ge 1$. Makes sense.
However- why can't I do this backwards? If I show the statement is true for $k - 1,$ aren't I showing that if the statement is true for $n = 1,$ it's likewise true for $n = 0, n = -1, n = -2, \ldots$
Also, why can't I prove the statement is true for $k + 0.1,$ and prove the statement true for $n = 1.1, 1.2, 1.3, \ldots?$ Both of these scenarios, in my mind, seem to follow the same logic as the "proper" definition of mathematical induction- but apparently they're no-go. Can someone please explain why?
Edit: The consensus seems to be that yes, even though it's abnormal, induction as I've stated above is logically sound. Which raises the question- why has my math teacher said this is wrong? Is it as I suspect, where she didn't want me straying from the proper definition of $k + 1$ induction and possibly confusing myself (or losing points on the test), or is there something else that makes the above fundamentally flawed?