# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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### Deducing this inequality from Cauchy-Schwarz?

I need to prove that for all $n \in \mathbb{N}$ we have the inequality $$\sqrt{\sum_{k=1}^n (x_k - y_k)^2} \leq \sqrt{\sum_{k=1}^n (x_k - z_k)^2} + \sqrt{\sum_{k=1}^n (z_k - y_k)^2}.$$ The hint ...
1answer
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### Extension of Fourier transform to $L^2(\mathbb{R})$

We defined the fourier transform and it's inversion for the Schwartz class $S(\mathbb{R})$. Since $S(\mathbb{R})$ is dense in $L^2(\mathbb{R})$, we can find for a given $f\in L^2(\mathbb{R})$ a ...
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### How to show that$\int_{0}^{\infty} \frac{sin(x)}{x}dx$ exists

How does one show that $\int_{0}^{\infty} \frac{\sin(x)}{x} \mathrm{d}x$ exists (i.e. does not equal $\infty$), with the most elementary methods possible?
1answer
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### Inequality involving ArcTan

How to prove that for $x\in[0, +\infty]$ the following inequality is true: $$\arctan x\geq\frac{3 x}{1+2\sqrt{1+x^2}}?$$ I don't have idea from where to start, so any hint is welcome. Thanks in ...
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### Real analysis reference for statistician

I'm a undergraduate statistics student, I think that learn Real Analysis can be useful to me in some points, can anyone suggest a introductory book for self-study ? I'm already multivariate calculus, ...
2answers
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### Closure of sets (specifically regarding the notation)

I'm new to sets and the notation is somewhat confusing to me. I just want to see if what I'm doing makes sense. For the following sets I need determine if it is open, closed, or neither. I also ...
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### Are all important function spaces vector spaces?

EDIT: I definitely agree with Mike Miller that the question as written originally/below is too general. I guess what I am trying to ask is that "does everything an analyst could ever care about have ...