# 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.

93,717 questions
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### Let $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$. Prove that $x_{n+1} < x_{n}$

Let $$x_{n+1} = \frac{1}{2}(x_{n} + \frac{a}{x_{n}})$$ Prove that $x_{n+1} < x_{n}$ for $a \geq 0$. Hint: Let the initial guess satisfy $x_{1} > \sqrt{a}$ I am stuck at how to begin this. I ...
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### Application of Jensen's Inequality to non-negative integrable function

I am reading a book, where it uses the following result. Can someone help me to derive the result? I know i have to use Jensen's inequality here, but not sure how to get the final result. Here is the ...
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### Multivariable limit of $\lim_{(x,y) \to (0,0)} \frac{2xy^4}{(x^2+y^2)^2}$

How can I calculate the multivariable limit of $\lim_{(x,y) \to (0,0)} \frac{2xy^4}{(x^2+y^2)^2}$? I'm new to this and I've seen a couple of examples, where it is possible to limit the fraction by a ...
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### How much do tails contribute to a Gaussian's total variance?

H${}$ello, if $X\sim \mathcal{N}(0,I_{n\times n})$ what is a good upper bound for $\frac{1}{n}\int_{A} \|X\|^2 d\mathbb{P}$ when $\mathbb{P}(A)<\varepsilon$? Thanks!
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