Any reasons why the basis case can't be at the end of a mathematical induction proof? When doing a proof by mathematical induction, I was wondering if there is any logical reason why the assumption (n=k) and induction (n=k+1) steps couldn't be done first, then do the basis case (n=1) afterwards, rather than the traditional way of doing this basis case first? I am teaching this to my students and I think it would make more sense to them to first show that P(k+1) is true if P(k) is true, and then show that P(1) is true, then P(2) must be true, P(3) is true, etc. and P(n) is true for all n. Anyway, I can't see any flaws in the logic? Please advise.
 A: There is pedagogical value in showing a proof by induction following in parallel the definition of proof by induction.  The definition tends to be something like "If $P(1)$ and if for all $k$, $P(k) \implies P(k+1)$, then for all $k$, $P(k)$."  Following this pattern helps students see that you actually are implementing the definition.
Once one is done with learning how to do induction, one is writing inductions in service of communicating a result to a reader.
A good way to organize a multi-component proof is Notation, Trivialities, Work.  The reason you handle trivialities first is that these are things the reader should be looking for, but which are disposed of quickly.  It most rapidly reduces the reader's cognitive load since they only have to remember that they are looking for a smaller list of results to complete the proof.  Consequently, if your base case is easier, put it first.  If your inductive step is easier, put it first.  Either order is logically adequate.
A: Formally, there is no requirement to prove P(1) first. Practically, though, verifying P(1) first can save a lot of aggravation in cases where the inductive step works, but the base step turns out to not hold true. Try to prove that $2n+1$ is even for $\forall n \in \mathbb{N}$ by induction, for example: the inductive step certainly works, but the proposition is false since the base case $2 \cdot 1 + 1$ turns out to be odd - and you'd only realize that at the very end if you were to check P(1) last.
(And then, there are those funny cases like All horses are the same color where the fallacy falls squarely in between the base step and the proper induction step.)
A: One way to make logic simpler to understand and write is to write things in their order of dependence.
Facts proven about P(1) can be used later on in the proof of P(k)->P(k+1) easily.
Facts proven about P(k)->P(k+1) being used to prove P(1) would be more suspect, because those facts rely on an assumption that P(0) is true when reasoning about P(1).
By putting the base case first, the argument with the fewest assumptions, it becomes easier to "cherry pick" things you proved when proving the base case "blindly".
Then you follow up with a case with more assumptions -- that P(k) is true for some k -- and proceed to prove P(k+1).
None of these matter in a formal sense, because the things proven during P(k)->P(k+1) that rely on P(k) have that requirement regardless of if it is written first.  But the idea that you start a logical proof and can "steal results" you proved earlier on for use later in the proof is a cognitive shortcut (that doesn't always hold).
A: Of course this is possible, there is no reason to do the base step first, it is however true that the fundamental idea of proof by mathematical induction is that

Let $A$ be a set such that $A\subseteq\mathbb N$, then let
$$1\in A$$ $$k\in A$$ $$k+1\in A$$
Hence, it can be inferred that $A=\mathbb N$, since $k$ can be an arbitrarily large value, and will still be an element of the set of all natural numbers.

Meaning naturally, any theorem that can be proven by means of mathematical induction is proven for $1$ first, and then for $k+1$. In addition, it is often easier to prove the desired theorem for $1$ than it is to prove the desired theorem for $k+1$, but of course even this is up to preference.
Ultimately, it is not written that proof by induction has to start with the base step, but note that if the theorem can be proven for the inductive step but not for the base step then the theorem is disproven.
A: Not only there is no reason, from the logical point of view, why the base case must be proved first as furthermore the very first text dealing with Mathematical Induction — Levi ben Gershon's Maaseh Hoshev (The Art of Calculation), written in 1321 — dealt with it doing first the inductive step and only after the base case.
A: I'm surprised nobody mentioned the computational aspect of this.
Many computer programs rely on recursive functions, which are effectively algorithmic implementations based on the same steps as an inductive proof.
In computer programs however, the base case(s) are typically dealt with first, and the 'recursive' step (i.e. essentially the "inductive hypothesis") is dealt with only if the base cases do not apply.
Since inductive proofs share such similarities with their algorithmic recursive implementations, notationally speaking it makes sense to be consistent in putting base cases first.
See http://jeffe.cs.illinois.edu/teaching/algorithms/notes/98-induction.pdf   (pages 6-8) for a nice example of this.
A: It is good practice, at all levels of explanation, not to make the reader read something that seems pointless.
Frequently the basis case is trivial and obvious, and there is no point wasting any time on it on its own. For instance, when proving "At any party there are at least two people who have the same number of friends", making a great fuss about "When two people are alone in a room, they both know the same number of people in the room" is pointless and ridiculous without knowing the induction that is coming. It is much more motivating to get the induction done first, so that the reader or listener knows "I am on the verge of proving infinitely many facts at once, if only I have a place to start". Once that is established, the "place to start" has a reason for existing, and there is a motivation to discover it and get one's head round it.
This all applies even more when (as sometimes happens) once can take the induction all the way back to $N=0$, which is practically meaningless as well as trivial.
Mathematical proof, like computer programming, is primarily a literary activity and whatever makes it work as good literature is what ought to be done. 
