# Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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### Prove that the sequence $\{n^4\}_n$ diverges to infinity.

My question is how would you go about proving this problem? Proofs are not my strong suit as I prefer solving for a specific answer. Anyway I read somewhere that you would prove the reciprocal ...
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### Is there a way to add a node to a digraph without adding a new topological sort?

I have a digraph with $n$ nodes. I want to add a new node to the graph, but I don't necessarily want to add a new edge to connect this node. In essence I want the graph to remain acyclic. Can I add ...
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### Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. Prove that there exists $c ∈ \left[0,\frac{1}{2}\right]$ such that $f(c) = f(c+\frac{1}{2})$.

Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. Prove that there exists $c ∈ \left[0,\frac{1}{2}\right]$ such that $f(c) = f\left(c+\frac{1}{2}\right)$. So, I know I'm supposed to use the ...
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### Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem.

Prove that there exists a real number $x$ such that $x^{177} + \frac{165}{1+x^8+\sin^2(x)} = 125$ using Intermediate Value Theorem. Uhhh I have no idea where to even start with this. Anything to give ...
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### Show that $e^{\alpha t} \ast (\frac{t^k}{k!}e^{\alpha t}) = \frac{t^{k+1}}{(k + 1)!}e^{\alpha t}$ for the convolution

As it says in the title, I would like to show that $e^{\alpha t} \ast (\frac{t^k}{k!}e^{\alpha t}) = \frac{t^{k+1}}{(k + 1)!}e^{\alpha t}$ for the convolution. However the only thing I know for the ...
72 views

### The set of all rational points in the plane is a countable set

From Kolmogorov's Introductory Real Analysis. I am doing some self-study and would like some feedback on whether my proof is correct. I am using that the set of rational numbers is countable as given, ...
### Use the definition of continuity to prove that $g$ is continuous at $x = 0$
Let $D$ be a subset of R containing $0$, and let $f : D → R$ be bounded on $D$ (i.e., $f(D)$ is a bounded subset of R). Define a new function $g : D → R$ by $g(x) = xf(x)$. (a) Use the definition of ...