# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with 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.

8 questions
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### Sobolev embedding for the $L^q$ norm.

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
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### $f(x)/x \to l$ and $f''(x) = O(1/x)$

$f \in C^2(\mathbb{R}, \mathbb{R})$ such that $f(x)/x \to l \in \mathbb{R}$ as $x \to + \infty$, and such that $f''(x) = O(1/x)$ at $+\infty$. Find : $$\lim_{x \to +\infty} f'(x)$$ Some thoughts : ...
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### Questions on the real bump function and conjuction of smooth functions

Before I ask you my question which I will mark in bold I will tell you what I already gathered so far. In a previous result I have showed that the bumpfunction is smooth. The bumpfunction is defined ...
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### Proof (without use of differential calculus) that $e^{\sqrt{x}}$ is convex on $[1,+\infty)$.

Prove (without use of differentiation) that $f(x)=e^{\sqrt{x}}$ is convex on $[1,+\infty)$. Attempt. Function $x\mapsto e^x$ is convex and increasing, but $x\mapsto \sqrt{x}$ is concave, so we ...
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### Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$, does there exist an ...
### Evaluating $\lim_{n \to \infty} n\left(1-\frac{1}{e}\left(1+\frac{1}{n}\right)^{n} \right)$
$$\lim_{n \to \infty} n\bigg(1-\dfrac{1}{e}\bigg(1+\dfrac{1}{n}\bigg)^{n} \bigg)$$ If I write expansion of $\bigg(1+\dfrac{1}{n}\bigg)^{n}$ it was equal to expansion of $e$ so $n-n=0$. Is limit is ...