Why is it wrong to start the inductive step with p(k)?

Question

I've never thought of this problem when I work with induction before. I am following this video lecture and the instructor mentions that in inductive steps it is wrong to start with p(k) and try to construct p(k+1). Specifically, it is mentioned that when trying to prove that every non-empty finite set of real numbers contain its supremum, it is wrong to start inductive steps with a k-element set and add one element to it while the correct approach is to start with a set of k+1 elements and find a k-element set within it. Here is the proof. Why is it wrong?

Answer 1

Induction always "starts" with $p(k)$. You assume $p(k)$ and try to prove $p(k + 1)$. I believe that you are confused about where to start the proof of $p(k + 1)$. This proof concerns arbitrary sets with $k + 1$ elements, so surely it must begin, "Let $A$ be a set with $k + 1$ elements; we will show that $A$ has a maximum."

You could start with a set containing $k$ elements, but eventually you must show that every set containing $k + 1$ elements is a set with $k$ elements plus an extra. These are just small semantics.

Explicitly, it is not wrong to start the proof of $p(k + 1)$ with a $k$-element set, it just may not be very helpful.