Difference between "Show" and "Prove" In many mathematics problems you see the phrase "prove that..." or "show that..." something is. What's the difference between these two phrases? Is "showing" something different from "proving" something in mathematics?
Thanks.
 A: No difference. Just another way of stating a question.
A: In Engelking's book "general topology" he uses "show" for easy proofs/examples, and "prove" for the hard ones. He also uses "note" (in exercises) for basically two line observations, often just for later reference (as exercises are an integral part of the text). He does explain this in advance, by no means this is standard. I do think it's convenient. It's a bit like Knuth numerically rating exercises in AoP. 
A: As noted above, they can be used interchangeable with "show" generally being used for more straightforward proofs while "proof" for more abstract or difficult ones. They are often times used to separate existential versus universal conditions. That is an author is more likely to say "show there is a bounded real-valued function on $\mathbb{R}$ that is not Lebesgue measurable" versus "prove that for all quadratic polynomials $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$".
Again, the words "show" and "prove" are interchangeable ("prove there is a ..." or "show that for all..."), but they are commonly used in that way. Or at least one or more of the author's I'm using now uses them that way.
A: Sometimes students misinterpret show to mean give an example. I now avoid using show in exams; I always use prove when a proof is required.
In the context of examples or calculations, it might be ok use show. For instance, "Show that $2$ is a root of $x^2-4$" or "Show that $\sin x$ is a solution of $y''= -y$.
A: No difference. Just a way to fool students...
A: I remembered when I was in high school my maths teacher said specifically there was a difference between showing and proving.
He said at our - high school - level, everything is generally considered a 'show' which are typically quite specific and do not involve abstract theories; whereas once you hit the higher levels, then you'd start with proofs which are considerably more rigorous. e.g. prove that 1+1 = 2.
A: To show is to use numbers to demonstrate a certain property, to prove it to use letters to prove that the system works for all number systems. There is a clear difference that can really hurt you in exam questions and assignments. Trust me, I know.
