# Is this a valid induction proof example ?

Learning induction proof now, found a "simple" example, which is a bit confusing to me (not sure if it is a valid example). If so, why the IH( suppose a root of rank k has at least $$2^k$$ vertices in its tree )is a valid one since it just repeating the statement that we are trying to prove ? What about the induction step?(Then a root ... in its tree.)

• Please do not include images containing text. All text should be included as such in the question; otherwise, it cannot be found using the site's search engine. Thank you. Dec 17, 2018 at 10:26

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 $$k$$.
• 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.