What are nice “proofs” of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples of what I have in mind:

$s=\sum_{i=0}^\infty \delta^i=1/(1-\delta)$.

“Proof:” $s=1+\delta+\delta^2+\cdots=1+\delta s$ and hence $s=1/(1-\delta)$.

$(f\circ g)'(x)=f'(g(x))g'(x)$.

“Proof:” $\displaystyle\lim\limits_{h\to 0}\frac{f(g(x+h))-f(g(x))}{h}=\lim\limits_{h\to 0}\bigg(\frac{f(g(x+h))-f(g(x))}{g(x+h)-g(x)}\frac{g(x+h)-g(x)}{h}\bigg).$

$[0,1]$ is uncountable.

“Proof:” Pick a number from $[0,1]$ randomly. Every number has the same probability. If this probability were positive, there would be finitely many such numbers such that the probability of picking one of them exceeds $1$, which cannot be. So the probability of picking each number is $0$. If $[0,1]$ were countable, the probability of picking any real number would be $0=0+0+0+\cdots$. But by picking from a uniform distribution, I will get a real number with certainty.

It might be helpful to indicate where the lapses in rigor are and why the method works anyways.

**Answer**

**Cayley-Hamilton Theorem.** Let $A$ be an $n\times n$ matrix, and let $f(t)$ be its characteristic polynomial. Then $f(A)=0$.

“*Proof*.” $f(t) = \det(A-tI)$. Therefore, $f(A) = \det(A-AI) = \det(A-A) = \det(0) = 0$.

*Issues.* One problem may not be obvious… the equation “$f(A)=0$” is really saying that the matrix we get via the evaluation (by identifying the underlying field with the subring of scalar matrices) is the zero *matrix.*

However, the “proof” claims to prove that $f(A)$, which is supposed to be a matrix, is equal to the value of a determinant, which is a *scalar*.

As to why it “gives the right answers”… well, because the theorem *is* true. I am reminded of what Hendrik Lenstra once said in class after presenting an idea for a proof and explaining why it didn’t quite work:

The problem with incorrect proofs to correct statements is that it is hard to come up with a counterexample.

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