# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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### Prove that the function $f:[2,4]\rightarrow\mathbb{R}$ is integrable.

Define $$f(x)=\begin{cases}x\qquad\text{if 2\leq x\leq 3}\\ 2\qquad\text{if 3<x\leq4}\end{cases}$$ Prove that the function $f:[2,4]\rightarrow\mathbb{R}$ is integrable. Let $f=f_1+f_2$ ...
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### “limit-truncation” argument in Donsker's theorem

I'm working out the proof of Donsker's theorem given in Revuz and Yor, Continuous martingales and BM, (theorem 1.9, ch. XIII, p. 518). At one certain point, they prove the following fact : for every ...
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### Determining the exact one from all possible Jordan Canonical Forms of a matrix

Here is the example I encountered : A matrix $\ M\$ $(5\times 5)$ is given and its minimal polynomial is determined to be $(x-2)^3.$ So considering the two possible sets of elementary ...
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### Integrate $\int \frac {\ln x}{x^2}$ by U-Substitution

I've got a question regarding "improper" u-substitution integrals. How would we solve $\int \frac {\ln(x)}{x^2}dx$ using the u-substitution $u = \ln(x)$? I know that it might be easier to solve it ...
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### Mistake in a proof concerning centers of incircles

In the above diagram we have that $D, E, F$ are the points of contact of the incircle of an acute-angled triangle $\Delta ABC$ with $BC, CA, AB$ respectively ,and $I_1,I_2,I_3$ are the incentres of ...
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### Proof that product topology of subspace is same as induced product topology

Let's assume that $A\subseteq X$ is product of $A_{i}\subseteq X_{i}(i\in I)$. Then product topology of $A$ is the same than the topology induced by $X$. I have proved this few different times now, ...
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### Putnam 2009 A4 clarification

Can anyone verify if this proof of the 2009 A4 Putnam problem is correct? Thanks! 2009 A4. Let $S$ be a set of rational numbers such that I. $0 \in S$. II. If $x \in S$, then $x \pm 1 \in S$. III. ...
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### Example that in general $|abc|\neq |cba|$

I tried to provide an example of group elements $a,b,c$ that shows that in general $|abc|\neq |cba|$. Please could you tell me if my example is correct? Let $G$ be the dihedral group $D_6$ (the ...
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### prove that $n(n+1)$ is even using induction

the base case of $n=1$ gives us $2$ which is even. assuming $n=k$ is true, $n=(k+1)$ gives us $k^2 +2k +k +2$ while $k(k+1) + (k+1)$ gives us $k^2+2k+1$ whats is the next step to prove this by ...
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### Definition of a vector of rational coordinates

What is a vector of rational coordinates? Consider a random vector $X$ of dimension $k\times 1$ taking values in $\mathbb{R}^k$. Then take $\mathbb{Q}_k:=(q_1,q_2,...)$ as the vectors with rational ...
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### The countable dense subset of every compact metric space

Show that any compact metric space has a countable dense subset. I am having problem with finishing the proof after a few steps. This is how I am going : So, let $X$ be the compact ...
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### Some doubts about easy computations involving nontrivial zeros of Riemann's zeta function

On assumption of Riemann hypothesis when I write a complex zero (nontrivial zero) of zeta function as $\rho=\frac{1}{2}+it_\rho$, and I write $x^\rho$ as $\sqrt{x}e^{it_\rho \log x}$, then multiplying ...
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### If $f$ is continuous and positive and $\int_a^b f(x)dx=0$ therefore $f(x)=0$ for all $x\in[a,b]$. [duplicate]

If $f$ is continuous and non-negative and $\int_a^b f(x)dx=0$ therefore $f(x)=0$ for all $x\in[a,b]$. My attempts Suppose $f\neq 0$, i.e. there is an $x\in ]a,b[$ s.t. $f(x)\neq 0$. Since $f$ is ...
Let X and Y be two Hilbert spaces. Let $f : X \rightarrow ]-\infty, +\infty]$ be convex and lower semicontinuous, let $A : X \rightarrow Y$ be linear, and let $g: Y \rightarrow ]-\infty,+\infty]$ be ...