# Questions tagged [solution-verification]

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### Find $g(x)$ if $f(x)= \begin{cases} -1, & -2 \leq x \leq 0 \\ x-1, & 0 < x \leq 2 \end{cases}$ and $g(x) = |f(x)| + f(|x|)$

$$f:[-2,2] \rightarrow \Bbb R$$ $$\text {and }f(x)= \begin{cases} -1, & -2 \leq x \leq 0 \\ x-1, & 0 < x \leq 2 \end{cases}$$ And, let $g(x)$ be equal to $|f(x)|+f(|x|)$ We need to find ...
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### Finding a bijective correspondence between $X^{\omega}$ and $\mathcal{P}(\mathbb{Z}_+)$

Let $X = \{ 0,1 \}$ and let $\mathcal{P} (\mathbb{Z}_+)$. Find a bijective correspondence between $\mathcal{P} (\mathbb{Z}_+)$ and the cartesian product $X^{\omega}$ or ${\bf show}$ there isn't ...
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### Prove that a sequence of complex numbers $z_{n} = (a_{n},b_{n})$ converges to $z = (a,b)$ iff $a_{n}$ converges to $a$ and $b_{n}$ converges to $b$

The complex numbers $\textbf{C}$ with the distance $d(w,z) = |w - z|$ form a metric space. If $(z_{n})_{n=0}^{\infty} = (a_{n},b_{n})_{n=0}^{\infty}$ is a sequence of complex numbers, and $z = (a,b)$ ...
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### Substitution for equation in two variables

Lets say I start with an equation $$3(1+x+x^2)(1+y+y^2)(1+z+z^2)=4x^2y^2z^2-1$$ and I get a solution in positive integers $(4,4^3,4^6)$. To go about showing that there are no solutions possible in odd ...
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### Proof check: Subset of separable space is separable

I want to know if my proof is correct and, if so, if it could be simpler. I think the last two lines aren't fully convincing. Let $X$ be a separable metric space. Prove that every subset of $X$ is ...
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### Given a sequence $x_{n}$ in the metric space $(X,d)$, prove that $L$ is a limit point iff there is a subsequence $x_{f(n)}$ which converges to $L$.

Let $(x_{n})_{n=0}^{\infty}$ be a sequence of points in a metric space $(X,d)$, and let $L\in X$. Then the following are equivalent (a) $L$ is a limit point of $(x_{n})_{n=1}^{\infty}$ (b) There ...
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### How do I show that if $p_n q_{n-1} - p_{n-1}q_n = 1$, then $p_n/q_n$ converges?

Let $p_{n}$ and $q_{n}$ be strictly increasing, integer valued, sequences. Show that $$p_{n}q_{n-1}-p_{n-1}q_{n}=1,$$ for each integer $n \ge 1$, then the sequence of quotients $\frac{p_{n}}{q_{n}}$ ...