3 votes
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Prove. If $r$ and $s$ are bisquare, then $rs$ is bisquare

Note that: $$(a^2+b^2)(x^2+y^2)=a^2x^2+a^2y^2+b^2x^2+b^2y^2$$ We would like to write this as sum of squares. Let's try: $$a^2x^2+b^2y^2=a^2x^2+b^2y^2+2abxy-2abxy=(ax+by)^2-2abxy$$ $$a^2y^2+b^2x^2=a^2y^...
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3 votes
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Injectivity of measures in measure theory

Here is a counter example. Consider $X = \{1,2,3\}$ and $\mathcal{A} = \{\emptyset, \{1\}, \{2,3\}, \{1,2,3\}\}$. This is a $\sigma$-algebra. Now define $\mu(A) = |A|$. This is injective. The issue ...
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2 votes
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Proof that function is differentiable using definition

Proof $2$ is correct, albeit missing precise explanation. The key point is that $|f(x)|\le x^2$ holds regardless of whether or not $x$ is rational. Proof $1$ has the right spirit (!) but is incorrect ...
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2 votes
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The Gram-Schmidt process: on the equality of the generated subspace.

For your inductive step, you have that $$ e_{k+1}=\beta_{k+1}x_{k+1}-\sum_{j=1}^k\beta_je_j $$ for certain coefficients $\beta_1,\ldots,\beta_{k+1}$. The inductive hypothesis gives you that $$ \sum_{j=...
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2 votes
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Question re: Collatz Conjecture

If we knew that every Collatz orbit eventually hits a number less than $10$ (is that what you mean?) then we'd be done immediately. So we don't know that! More generally if we knew that every Collatz ...
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2 votes
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Let $x$ be a fixed real number. Prove: if $x^5 + 2x^3 + x < 50$, then $x < 2$

Note that $2^5+2\times2^3+2=50$, so for all $x>2$ you have $x^5+2\times x^3+x=50$. Can you conclude?
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1 vote
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Is there a way to define universal quantification in such a way, that I can prove$\forall xPx\land Q\iff \forall x(Px\land Q)$ and similar statements?

You can prove this equivalence this way over a finite domain, but there are other ways to this type of equivalence that work for infinite domains and are easier. I actually don't know whether the ...
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  • 7,200
1 vote

Theorem 6, Section 2.3 of Hoffman and Kunze’s Linear Algebra

I think is false: In ${\Bbb R}^3$ with canonical basis let $e_1+e_2,e_2$ be a basis for $U$ and $e_1+e_3,e_3$ be a basis for $W$. Then $U\cap W$ is generated by $e_1$ which is not among the given ...
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1 vote

Let $x$ be a fixed real number. Prove: if $x^5 + 2x^3 + x < 50$, then $x < 2$

You can find the zeroes of $x^5 + 2x^3 + x$ using your factorization. For $x^4 + 2x^2 + 1$, you can solve $u^2 + 2u + 1 = 0$, where $u = x^2$. Once you solve for all the zeroes, you know that a ...
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1 vote

Let $x$ be a fixed real number. Prove: if $x^5 + 2x^3 + x < 50$, then $x < 2$

Let $f(x) = x^5 + 2x^3 + x$ So, $\frac{df}{dx} = 5x^4 + 6x^2 + 1$ Clearly, $\frac{df}{dx} > 0 \;\; \forall x \in \mathbb{R}$. So the function is strictly increasing. This means if $f(x_0) = c$ for ...
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