This is a perfect example of a sloppily written induction proof. Of course the induction hypothesis is not only repeating the proof's statement; otherwise, it would not qualify as a proof at all since it would be assuming the statement is true regardless of the ensuing argument. What is meant, instead, is that $k$ is then fixed and the induction step is conducted for this particular choice of $k$.
The following proof structure is more elegant and IMO much easier to follow (esp. for beginners):
- Induction basis: [...]
- Induction step: let $k$ be given.
Induction hypothesis (IH): let [statement you are trying to prove] be true for $k$.
Then [ensuing reasoning which shows the statement is true for $k + 1$ by making use of IH].
This makes it explicit for which $k$ the hypothesis should hold as well as what statement it is exactly that we are trying to prove in the induction step.