# Tag Info

### Proof that there are infinitely many primes congruent to 3 modulo 4

Does this work? This is a modification of Euclid’s proof. Let $p_1,p_2,…,p_k$ be the only 3 mod 4 primes. Consider two cases : (I) $k$ is odd Then notice $p_1p_2…p_k+4$ is a 3 mod 4 number, and in ...

### How to prove the product of first n consecutive odd numbers is a square less than another square?

You might note that the difference sequence of the sequence of squares is exactly the odd numbers. Here's the sequence of squares: $$0, 1, 4, 9, 16, 25, 36, \ldots$$ Here's the sequence of the ...
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### Exercise 7(a), Section 31 of Munkres’ Topology

Edit: I am expanding this instead of commenting since stack exchange doesn’t seem to like so many comments: Approach 2 does not work. In order to use your cited theorem, you would need the following 1 ...
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### Why Can't $\delta$ depend on $x$ in $\delta-\epsilon$ Proofs

You are trying to find a single $\delta$ value ahead of time that works for many cases. The cases are where $0 < |x-a| < \delta$, and the definition involves trying to find one delta $\delta$ ...
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### How to prove the product of first n consecutive odd numbers is a square less than another square?

Note that any product of two distinct odd numbers or two distinct even numbers can be written as a difference of squares. Let $p<q$ be both odd or both even. Then $(q-p)/2$ and $(q+p)/2$ are ...
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### Is this valid to prove that $2 \ \vert \ x^3 \Rightarrow 2 \ \vert \ x$?

This method is not valid. You showed that $2(4k^3)\neq x^3$ for all $k\in\mathbb Z$ correctly, but but to show that $2\nmid x^3$, you need to show that $2n\neq x^3$ for all integers $n$, not just ...
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### Let $x \in (0,1).$ How to show that $x \in \bigcup \left(0, \frac{n}{n+1}\right)$?

Since $\frac{n}{n+1}$ is an increasing function with argument $n$ on $\mathbb{N}^+$, then we have \begin{align*} \bigcup_{n=1}\left(0,\frac{n}{n+1}\right)&=\left(0,\lim_{n\to \infty}\frac{n}{n+1}\...
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### How to write maths

Singular can be the instance while the plural can be the iteration or collection of instances although it would be contextually dependent. Multiplication - $x(n)$ Multiplications - $\int_{x(n)}$ In ...
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### Exercise 12, Section 30 of Munkres’ Topology

Both parts of your proof are correct. However, since you included a proof-writing tag in your question, I'd like to add a quick comment about style! You use a lot of symbols, and that makes your proof ...
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### Proof Verification for $e^x$ is not uniformly continuous on R

The $\delta$ you choose depends on the $\varepsilon$ so you can't take $\varepsilon$ in terms of $\delta$. However, if you fix $\varepsilon >0$ and suppose that there exists $\delta>0$ such that ...
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### Proof Verification for $e^x$ is not uniformly continuous on R
Choose the sequence $\log(n+1)$ and $\log n$ and this two are parallel sequences but $e^{\log(n+1)}$ and $e^{\log n}$, not to be a parallel sequences, so that's why $e^x$ is not uniformly continuous.