# Tagged Questions

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

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### Prove that the set of all binary sequences is uncountable

Question: Prove that the set of all infinite binary sequences is uncountable. Comments: I think that there are a couple of ways of going about this. My first approach was to show that the set of all ...
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### Prove by induction that $\sum_{i=1}^n i \geq \frac{n^2}{2}$ [closed]

Can someone show me a formal proof of this exercise ? $$\sum\limits_{i=1}^n i \geq \frac{n^2}{2}, \quad \forall n \in \mathbb{N}.$$ Thanks to anyone who can help! :)
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### Prove a relation of a distance function

I had to do an exercise with this function: $$d_M:\Bbb C \rightarrow \Bbb R, \quad z \rightarrow inf\{ |z-w|; w \in M|$$ with $\emptyset \neq M\subset \Bbb C$. First I proved that this function is ...
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### Looking to receive feedback on elementary proofs in topology

I'm looking to receive some feedback on a couple of proofs I wrote verifying the discrete and trivial topologies and another simple topology. I'm inexperienced with proof (in the sense that I haven't ...
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### Proving path length, transitive closure

Set A is finite with n elements. Suppose a and b are elements of a set A with a != b. Let R be a relation on the set A so that there is a path from a to b of length at least 1. Show there is a path ...
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### Prove that (0,1) ⊆ R and (4,10) ⊆ R have the same cardinality [duplicate]

Hows does (0,1) and (4,10) both existing as real numbers make it have the same cardinality?
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### Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field?

Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field ? I am claim that it is not. My reasoning is this... What I am ...
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### Is this a valid proof of set membership

Let $S=\{x \in\mathbb{Z}: x\geq0, x=b-a ×m$ for some $m\in\mathbb{Z}\}$. Prove that if $b\geq0$ then $b$ is an element of $S$. Pf: suppose $b\geq 0$ Let $a$ be an integer define $b=b-a×m$ Where ...
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### Explanation for the the number of trailing zeros in a factorial.

I was doing a programming problem that asked that I find the number of trailing zeros for a factorial, and I came up with this: ...
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### On the limit of $f(n)$, specifically having to do with integration of an iterated $\arctan$

Assume we are given that $A_n(x)$ denotes $n$ iterations of $\arctan(x)$, for example $A_2(x)=\arctan (\arctan(x))$ If $$f(n)=\int_{0}^n A_n(x)\space \text{d}x$$ I am looking for a rigorous proof ...
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