Tagged Questions

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

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How to integrate $\int_{-3}^3 (x^2-3)^{3} \,dx$ without expanding the polynomial?

How can I integrate: $$\int_{-3}^3 (x^2-3)^{3} \,dx,$$ neither expanding the polynomial nor using the relationship between integral and derivatives? I mean, there is a way to compute this integral ...
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Proof writing: $\sum_{n=1}^{\infty}| a_n|<\infty$ implies $\sum_{n=1}^{\infty} a_n^2<\infty$.

Let $\sum_{n=1}^{\infty} a_n$ be an absolutely converging series. By definition, this means $\sum_{n=1}^{\infty} \lvert a_n\rvert$ converges. We want to show that $\sum_{n=1}^{\infty} a^2_n$ ...
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Show $\mathbb{N}^{\{0,1\}}$ is uncountable with a hint

Let $\mathbb{N}^{\{0,1\}} :=\{f: \mathbb{N} \to \{0,1\}\}$ is uncountable I have never heard of the table approach, and all the proofs say uncountability of $\mathcal {P}(\mathbb{N})$ I have seen so ...
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Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric.

How can i proof the following statement: "Any n x n matrix A can be written as a sum A = B + C where B is symmetric and C is skew-symmetric." i tried to work out the properties of a matrix to be ...
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Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold.

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold. The boundary is defined by $z = \sqrt{x^2+y^2}$. I'm trying to think how to approach ...
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Prove $\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$

I know the definition of divides into is $$a|b \equiv \exists a\in\mathbb{Z}, b = ac$$ however I'm not sure how to manipulate this to prove $$\forall n \in\mathbb{N}, (n|105 \wedge n|70) \implies 5|n$$...
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Proving $n < 2^n$ by Cantor's theorem

So we know Cantor's Theorem is of course. For any set $S$, the power set $P(S)$ has a strictly greater cardinality, $\iff \#S < \#P(S)$. We seek to prove $n < 2^n$ using this information. I ...
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Proving the u-substitution formula

Let $g: [a, b] \rightarrow [c,d]$ be continuously differentiable and $f: [c,d] \rightarrow \mathbb{R}$ continuous. Prove that 􏰀$\int_{a}^{b} f(g(x))g'(x) dx$ = $\int_{g(a)}^{g(b)} f(t)dt$ ...
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Disprove: $f\circ g = f \circ h \implies g=h$ for a surjective function $f$

I tried using a very specific counterexample here where I select a surjective function for which the compositions are equal but the functions within are not. This is probably off-base, but it's what ...
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Prove $\forall n \geq 10, 2^n > n^3$

Prove $\forall n \geq 10, 2^n > n^3$ base case: $n = 10$ $2^{10} = 1024$ $10^3 = 1000$ $1024 > 1024$. So $P(k)$ holds for $k = n$. We seek to show $P(k+1)$ holds: We know $2^k > k^3$. ...
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Equivalence relation and class, Proof.

The relation $P$ on $ℝ$ is defined by $xPy$ iff $x^2=y^2.$ (a) Prove that the the relation $P$ is an equivalence relation. (b) Describe the equivalence class of $3$ and $0$. In order to ...
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Nullity and an Isomorphism

I'm working on some introductory proofs in linear algebra, and I think that I could use some help on this particular problem. I want to prove that a linear surjective map $T: R \rightarrow W$ is an ...
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Prove that $2^n$ does not divide $n!$

I want to prove that $2^n$ does not divide $n!$. I was trying by induction and I'm confused about if what I'm doing is right. First I test it with $n=1$. In fact: $$2^1 \nmid 1!$$ So if i take the ...
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For every integer $m\geq 0$ let $I_m=\int_0^1x^m\left(x^2 -1 \right)^5dx$. Prove that for $m\geq 2$ $I_m= \frac{m-1}{m+11}\,I_{m-2}.$

This is a very interesting problem that I came across in an old textbook of mine. So I know its got something to do with integrals, but other than that, the textbook gave no hints really and I'm ...
How to show the usual topology is finer than co-finite topology on $\mathbb{R}$
$\mathbb{Q}$ is countable, so we can list its elements. Let $\mathbb{Q} = \{r_1 , r_2, ...\}$ Define $f: \mathbb{R} \to \mathbb{R}$ by the following rule: \$ f(x) = \begin{cases} 0 & x \...