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 $n\geq 1$. 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 it 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?