# Tagged Questions

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

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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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### Prove: If $G$ and $H$ are disjoint $G_{\delta}$, then there exists an $F_{\sigma}$ set $B$ such that $H\subseteq B$ and $B\cap G=\emptyset$

Problem. Prove: If $G$ and $H$ are disjoint $G_{\delta}$ sets, then there exists an $F_{\sigma}$ set $B$ such that $H\subseteq B$ and $B\cap G=\emptyset$. Please HELP. Thank you..
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### Questions about proving by cases for a biconditional

Let's say I have a predicate, $\forall x\in h(x) : f(x) \leftrightarrow h(x) \lor g(x)$ I understand that when Q $\implies$ P that we do cases and assume both h(x) and g(x) in each case, however when ...
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### How to prove that $\forall n \in \Bbb{N}$, $n+\lfloor \sqrt{n} +\frac12 \rfloor$ cannot be a complete square?

Here is my thinking process: Let $k=4n$, so $n=\frac{k}{4}$, then $k\in \Bbb{N}$. Then the expression becomes $\frac{k}{4}+\lfloor \frac{1}{2}\sqrt{k} +\frac12 \rfloor$. But then I don't know how to ...
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### How to prove that $\forall x\in \Bbb{Q}:\ x\ne 0\implies [\exists a,\ b\in \Bbb{I}: x=a\cdot b]$ if $\Bbb{I}$ is set of irrational numbers?

I initially thought contrapositive would be easier, so I wrote $\forall x\in \Bbb{Q}:\ [\forall a,\ b\in \Bbb{I}: x\ne a\cdot b]$ $\implies x=0$. But I still had no idea how to start. Could someone ...
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### Show that the collection $\tau^*=\{(a,b]:a,b\in \mathbb{R},a<b\}$ is a basis for a topology in $\mathbb{R}$

Show that the collection $\tau^*=\{(a,b]:a,b\in \mathbb{R},a<b\}$ is a basis for a topology in $\mathbb{R}$.Help me on this.Thank you very much in advance..
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### Let $X$ be a topological space and $A\subseteq X$. Prove $\operatorname{Fr}(A)=\emptyset$ iff $A$ is both open and close in $X$.

Let $X$ be a topological space and $A\subseteq X$. Prove $\operatorname{Fr}(A)=\emptyset$ if and only if $A$ is both open and close in $X$. Can you help me on this?
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### Terminating each branch of a proof with $\square$

My question is a variation on this one. I have a proof which divides at the top level into a number of mutually exclusive cases, with further partitioning within that. Is it reasonable to place a ...
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### how to prove that invertible matrix and vectors span the same space?

Given $M$ is an invertible matrix, and {$\vec{v_1}...\vec{v_k}$} spans $R^n$, then {A$\vec{v_1}...A\vec{v_k}$} also spans $R^n$ What does matrix invertibility have to do with span?
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### What is the right way to show that $\lim\limits_{n \to \infty} f_n(x_n) = f(x), x_n \to x$

There is a question (could be from Ruddin's real analysis text) To show that $\lim\limits_{n \to \infty} f_n(x_n) = f(x)$, is equivalent to show that: $$|f_n(x_n) - f(x)| < \epsilon$$ Which ...
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### A newbie need some help in proof building. How to prove that any regular expression admits a disjunctive normal form?

Prove that any regular expression admits a Disjunctive Normal Form, i.e.: R = R1 U R2 U … Rn , where neither Ri contains a union. I would like some help with this question. If you could push me into ...
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### Deductive proof in natural numbers - division

Prove, using induction rule: $$\forall_{n\in N} \left (2^{2n+1} + 3n + 7 = 9c\right)$$ $$c\in N$$ 1. I checked with 1 : works 2. I assumed that it is true for some natural number k 3. I plugged in ...
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### $\gcd(3k+2,5k+3) = 1$ for all integers $k$

I'm doing this question for an assignment, the question is: Prove: if $k\ \epsilon\ \mathbb N$, then $gcd(3k+2, 5k+3)=1$. I was going to do it by induction, so what I have so far is: $n\ |\ 3k+2$ ...
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### How to prove $\forall x,y\in\mathbb{R}: x^2+y^2 = (x+y)^2 \Leftrightarrow x=0\lor y=0?$

The question I really have is the structure and I am not sure to use pack-unpack or not. Here is my try: Let $x,y\in\mathbb{R}$ Assume $x^2+y^2 = (x+y)^2$ Then $x^2+y^2 = x^2+2xy+y^2$ #by ...
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### Proving $x \lt y \iff x^n \lt y^n$

Let F be an ordered field. $x,y \in F, x,y \ge 0$ and $n \in N$ Prove that: $$x \lt y \iff x^n \lt y^n$$ Now I'm choosing to use induction to prove this (is that the only way to prove this?) ...
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### Proof of Thue's lemma

Here is the statement : "Let $m\in \mathbb{N}$, $a \in \mathbb{Z}$, if $\gcd(a,m)=1$ then there exists $x,\ y \in \mathbb{N}^*$ such as $x, y < \sqrt m$ and $ax \pm y \equiv 0 \ [m]$ ". I do not ...
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### Matrices Invertibility Existence

I have two square matrices, $A$ and $B$, and I know that $AB = I_n$, how do I then show that $A$ is invertible? I've been considering determinants, and I have started off by assuming that $A$ is not ...
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### Let $D$ be the open disc centred at $i$ and radius $3$. Prove that $|z-\omega|<6$ for all $z,\omega \in D$

I can see why it is less than $6$ because the longest distance is from either side through the origin and it can't be $6$ because it is an open disc. I think to prove formally I should use the ...
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### Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but …

Construct a sequence of measureable sets $E_1\supseteq E_2 \supseteq E_3 \supseteq \cdots$ such that $\mu(E_n)=\infty$ for each $n$ but $$\mu\left(\bigcap_{n=1}^\infty E_n\right)=0$$ Claim: Let ...