I have a general question concerning "rigor" when it comes to proof by induction. I make instructional videos about math, and I was in the process of planning a video on this topic. So I did what I always do before I make a video, which is check out other videos on the topic to see how other people are teaching / explaining it. What I found is that a lot of people are teaching the proof by induction method in a way that is (in my opinion) a lot less rigorous than the way I was taught. It just doesn't seem right.
So I figured I'd ask for a second opinion, since I'm sure some of y'all have a lot more experience with this than I do.
The way I was taught was: first show the basis step (show the statement is true when $n=1$). Then assume the statement is true when $n=k$, and USE THIS ASSUMPTION to show the statement is true when $n=k+1$.
This is where I see people going two different directions. Some people are doing it the way I was taught, which is that you have to work with the induction hypothesis to conclude the statement is true when $n=k+1$. Others however, are assuming the induction hypothesis, then writing something like "this is what we want to show:", then modifying what they want to show (the statement that $n=k+1$) until they arrive at something that is true (typically the induction hypothesis). Technically, I don't think they are making any extra assumptions, but for some reason it feels like they are working backwards and seems kind of sloppy. Am I right about this, or am I being too critical? I'm very interested to hear y'alls opinions. Thanks!