# Tagged Questions

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

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### Decomposition of a function into positive and negative parts and its integrability

1)Is it true that any function can be decomposed as a difference of its positive and its negative part as $f=f^{+}-f^{-}$ or that function should belong to $\mathcal{L}^{1}(\mathbb{R})$. Also if that ...
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### Definition of function

When defining a function, mathematicians often write something like this: Let $f\colon \mathbb R\to \mathbb R$ be the function given by $x\mapsto x^2$. The purpose of this definition may be to ...
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### How can I prove $\frac{m+a}{m+a+l+b}$ is between $\frac{m}{m+l}$ and $\frac{a}{a+b}$?

Let $m,a,l,b \in \mathbb{Z}^{+}$ How can I prove $\frac{m+a}{m+a+l+b}$ is between $\frac{m}{m+l}$ and $\frac{a}{a+b}$?
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### Continuous functions in the product topology on $\Bbb{R}^{\Bbb{N}}$

I'm trying to prove the following statement: Let $(X, T )$ be a topological space, and let $f : X \rightarrow \Bbb{R^{\Bbb{N}}}$ be a function, where $\Bbb{R^{\Bbb{N}}}$ has the product topology. Let ...
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### The Greatest Number of Edges on a Bipartite Graph

Let $G$ be a bipartite graph on $p$ vertices. Find a formula in terms of $p$ that determines the greatest number of edges that $G$ could have. Prove that this formula is correct. Let $V$ be the set ...
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### Style guide/typeface for handwritten mathematics

When writing math on graph paper, it's a small struggle to make my work as legible as possible and also use the page as efficiently as possible. I've read a little online about how latex typesets, ...
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### How do I prove an algorithm has $n^3$ time complexity?

Take the CYK algorithm outlined here: How to prove CYK algorithm has $O(n^3)$ running time In the top answer, how did that person go from the three summations to $t=(n^3−n)/6$ ? What's the method ...
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### How to describe “the digits of n” mathematically where n is an integer?

Suppose n = 12345 The sum of the digits of n = 1 + 2 + 3 + 4 + 5 = 15 For example, in Python, we might isolate the digit 1 by writing n[0]. How would one represent the digits of n mathematically?
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### proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
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### Challenging Series example [duplicate]

Let $\{x_n\}$ be a decreasing sequence such that the series of $x_n$ converge. Show that the limit as $n$ approaches infinity of $\{nx_n\}$ equals zero.
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### Box topology is finer than the uniform topology on $\mathbb{R}^\mathbb{N}$

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that ...
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### How to construct a proof once we intuit a solution

For any integer N, there is an integer P such that one of the following is true: N = 10P N = 10P + 1 N = 10P + 2 N = 10P + 3 N = 10P + 4 N = 10P + 5 N = 10P + 6 N = 10P + 7 N = 10P + 8 N = ...
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### Uniform topology is finer than the product topology on $\mathbb{R}^\mathbb{N}$

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The ...
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### How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
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### Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
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### Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
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### Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
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### Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
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### Proving the Baire Category Theorem from scratch, Stuck!

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other ...
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### Are theorems like subroutines for math? [closed]

I've been developing more appetite for math just lately, as I study electromagnetics to deepen my understanding of electric circuits and devices. I'm finding that doing derivations as exercises helps ...
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### Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
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### proof of Triangle Removal Lemma

Where can I find a proof of the following version of Triangle Removal Lemma (or any version equal to it)? Let $G(V,E)$ be a graph on $n$ vertices such that it contains $\varepsilon n^3$ triangles, ...
### Proof by induction: inequality $n! > n^3$ for $n > 5$
I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
My attempt: I imagined that if two sets are equivalent there would exist $f:X→Y$ that is bijective. If I conceptually create P(X) and apply the function defined for the first equivalence relation to ...