# Tagged Questions

For questions about the formulation of a proof, not about the mathematics behind it.

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### What kind of proof is this?

Let's say that we want to prove that object A is blue. Is the following reasoning true? First assume that $A$ is indeed blue. Then, use other axioms to show that depending on a control parameter ...
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### Proving a basis and dimension

Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose $C_{ij}$ is the $2\times3$ matrix with $1$ in the $i,j^{th}$ entry and zero ...
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### How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
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### How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
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### Clean proof for showing $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable

I wish to show that for $f:\mathbb{R} \to \mathbb{R}$, $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable Looks fairly easy if $f^{-1}([a,b))$ is one piece. Suppose ...
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### How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
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### Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
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### Proof using induction: $n! > n^2$, for $n\geq4$

Proof using induction: $n! > n^2$, for $n\geq4$ Basis: For n = 4, we have: $4! > 4^2$ $24 > 16$ (TRUE) Inductive step: By the induction hypothesis: $k! > k^2$ $(k+1)k! > (k+1)k^2$ ...
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### Proof help: Prove that $x^2+y^2+z^2 \geq xy+xz+yz$ [duplicate]

$x^2+y^2+z^2 \geq xy+xz+yz$ for all real numbers, x, y, and z. I'm not very good with working inequality proofs. Can someone help me prove this? The technique doesn't really matter.
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### Existence of convergent sequence in a closed set?

Is it true that if $C$ is a closed set, and $x \in {C}$ then there exist a sequence $\{x_i\}$ which tends to $x$? I'm not sure about the correctness of this statement, and whether it is true for ...
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### I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are ...
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### Prove that graph with odd number of odd degree vertices does not exist

I need to prove that it is impossible to have a graph in which there are an odd number of odd degree vertices. What is the easiest way to formally prove this? I feel that I can prove it just by ...
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### Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$. It was suggested to me that the easiest way to approach this was with a proof by ...
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### Prove that $3^x$ divides $(3x)!\,\,\,\, \forall x \ge 1$ [closed]

Prove that $3^x$ divides $(3x)!\,\,\,\, \forall x \ge 1$ For example, $3$ divides $6 = 3!$
Can someone please help me with this proof? Proof by induction that for finite sets, A and B, an injection $f: A \rightarrow B$ exists if and only if A is finite and $|A| \le |B|$.