# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
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### Show that an entire function that is real only on the real axis has at most one zero, without the argument principle

Could someone advise me on how to approach this problem: Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero. without the use of argument principle ...
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### The differential $\text d F_p$ is injective iff the pullback $F_p^*$ is surjective.

I'm trying to prove the following claim: Let $F\colon M \to N$ be a differentiable application beetween $C^\infty$ manifolds. Then the differential $\text dF_p\colon T_p M \to T_{F(p)}N$ is ...
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### If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0.$

Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$ could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$ Thank you.
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### Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

Thanks for the help! I've written a more detailed proof. The hints were great.
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### Proof Techniques ( Soft Question )

I've been googling around for books of methods of mathematical proofing, and I haven't had much luck finding anything reputable in book form. I do recall running by a few in a university library ( I ...
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### $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite
I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...