Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, the least upper bound property; and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus. This tag can also be used for more advanced topics, like measure theory.

Real analysis is a branch of mathematical analysis, which deals with the real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

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102
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6 answers

Continuous bijection from $(0,1)$ to $[0,1]$

Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map.
101
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3 answers

Does convergence in $L^p$ imply convergence almost everywhere?

If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$?
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101
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4 answers

How did Hermite calculate $e^{\pi\sqrt{163}}$ in 1859?

Pretend you are in 1859. What is a fast, efficient, and accurate way to numerically evaluate constants like that to, say, 20 decimal places, using ONLY pen and paper?
Tito Piezas III
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99
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16 answers

If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$

I'm studying for qualifying exams and ran into this problem. Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim_{n \rightarrow \infty} n a_n = 0$. Using the definition of the…
dls
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99
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0 answers

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the Implicit Function Theorem (finite-dimensional vector…
98
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8 answers

Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$

Compute $$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
user 1591719
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97
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5 answers

Self-studying real analysis — Tao or Rudin?

The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here comes another one. Recently I've began to learn real analysis via Rudin. I would do all the exercises, and if I…
96
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15 answers

How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not enough to prove this theorem.
faceclean
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96
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8 answers

Are calculus and real analysis the same thing?

I guess this may seem stupid, but how calculus and real analysis are different from and related to each other? I tend to think they are the same because all I know is that the objects of both are real-valued functions defined on $\mathbb{R}^n$, and…
Tim
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96
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7 answers

Is the "determinant" that shows up accidental?

Consider the class of rational functions that are the result of dividing one linear function by another: $$\frac{a + bx}{c + dx}$$ One can easily compute that, for $\displaystyle x \neq \frac cd$ $$\frac{\mathrm d}{\mathrm dx}\left(\frac{a + bx}{c +…
95
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2 answers

If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why is it the case that $$\|f_k - f\|_{L^p} \to 0?$$ A…
94
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12 answers

References for multivariable calculus

Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it…
94
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8 answers

If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course. There is an proof in which I don't understand some parts of it. The proof relates to the following proposition: ($S$ - infinite $\sigma$-algebra on $X$) $\implies $ $S$ is…
Belgi
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93
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3 answers

How to evaluate $\int_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1} \,\mathrm{d}x$ using complex analysis?

We were told today by our teacher (I suppose to scare us) that in certain schools for physics in Soviet Russia there was as an entry examination the following integral given $$\int\limits_{0}^{\infty} \frac{x^{-\mathfrak{i}a}}{x^2+bx+1}…
user505183
89
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6 answers

Contest problem: Show $\sum_{n = 1}^\infty \frac{n^2a_n}{(a_1+\cdots+a_n)^2}<\infty$ s.t., $a_i>0$, $\sum_{n = 1}^\infty \frac{1}{a_n}<\infty$

The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a sequence of positive real numbers and the series $\displaystyle\sum_{n = 1}^\infty \frac{1}{a_n}$ converges. Show that…
Potato
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