Questions tagged [real-analysis]
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
150,452 questions
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How fast can I guarantee convergence on a compact interval?
Suppose we have a sequence $(x_n)_n\subset [a,b]$, we know by compactness there must be at least some limit point, i.e. there exists a subsequence $n_k$ and $x\in[a,b]$ such that
$$x_{n_k}\xrightarrow{...
- 5,554
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Behaviour of sequence defined by recurrence
I am a bit rusty on the topic, and I have been presented the following exercise I am not really sure how to deal with. We have the sequence
$$ \begin{cases} x_0 \geq 0 \\ x_{n+1} = 2|x_n -1| \end{...
- 3,764
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Prove $\int (f - x/2)^2 \le a^3/12$ given $(\int f)^2 \ge \int f^3$ [duplicate]
I am working on a challenging integral inequality problem. I would appreciate a fully rigorous proof, especially concerning how to deduce the necessary pointwise constraints on the function $f(x)$ ...
-2
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0
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How to find the maximum value of $g(x)=|f(x)−h(x)|$ where $f(x)$ is a finite geometric series? [closed]
I was playing around with functions and got stuck trying to find the maximum value of a function.
I started with $f(x)$ and $h(x)$, where $h(x)$ is the result after applying the sum of geometric ...
- 42
1
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1
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Treating derivative like a number [duplicate]
This question stems from this one: A very weird integral equation
In the fourth step op uses the differential operator like a constant number and factors $f(x)$ out of the equation
$\frac{df}{dx} -f(x)...
0
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0
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How to prove a recursively defined sequence converges to 2 [duplicate]
Question 3.37 from Jay Cummings' Real Analysis asks the reader to prove that the sequence defined recursively by
$$
a_{n+1} = \sqrt{2 + a_n}\,,
$$
with $a_1 = \sqrt{2}$, converges to 2.
Proving that ...
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2
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How to bound $\mu(|x| > c)$ for a standard Gaussian measure $\mu$ on $\mathbb{R}^n$
Let $I$ be the characteristic function
$$I(x) = \begin{cases} 1 \quad |x| \leq c \\ 0 \quad |x| > c\end{cases}$$
and let $\mu$ be a standard Gaussian measure on $\mathbb{R}^n$.
How can I prove that
...
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0
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0
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13
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Quasi-analytic classes are dense in its dual? Analytic functions are dense in the hyperfunctions?
Let $X$ be a compact real-analytic manifold. I know that by adding an extra condition to a Komatsu-Romieu sequence $\mathscr{M}$ one obtains the nonquasi-analytic class, and that this additional ...
0
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1
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Let$\ \ f:[0,1]\to \mathbb{R}$ be a continuous function. Prove $\ \ \lim_{\lambda\to\infty}\int_0^1 f(x)\sin(\lambda x)\,dx = 0$. [duplicate]
NOTE- The source of question is Advanced Calculus on real axis which doesn't cover $\textsf{Lebesgue integral}$ and $\textsf{Measure Theory}$ so therefore an approach by those wouldn't be of much ...
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1
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Writing a sequence in $c_0$ as the product of two sequences with one in $l^p$
Question. Can I write any sequence $(\mu_n)_n \in c_0$ as the products of two sequences with one of them in $l^p$? That is, can every element of $c_0$ be written as $\mu_n = a_n b_n$ such that say $(...
- 179
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2
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Showing that $\nexists r \in \mathbb{Q} : 2^r=3$
I am currently self studying real analysis from the book Understanding Analysis, Stephen Abbott, 2nd edition. In page 11, exercise 1.2.2 the problem asks to show that there is no rational $r$ ...
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1
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Principle of Recursive Definition (Royden 3rd ed. 1988)
I am trying to flesh out a formal proof for the Principle of Recursive Definition as stated in Royden, 3rd edition.
Principle of Recursive Definition: Let f be a function from a set $X$ to itself, ...
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1
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1
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Does uniform convergence to zero function on every closed and bounded interval imply uniform convergence on $\mathbb{R}?$ [duplicate]
I am stuck to the following question while self studying Real Analysis.
Let $f_n: \mathbb{R} \rightarrow \mathbb{R},$ and suppose that $f_n \rightrightarrows 0 \ (f_n$ converges uniformly to the zero ...
- 53
5
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2
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One doubt on the proof that the Fourier transform of a function is in $C_0(\mathbb R)$
One proof of the statement that the image of Fourier transform $F: L^1(\mathbb R)\to L^{\infty} (\mathbb R)$ is contained in $C_0(\mathbb R)$ is as follows:
First we use straight-forward computation ...
- 2,892
4
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1
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296
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Find the integral over a unit disk
I'm trying to solve the following problem:
Let $B=\{(x,y):x^2+y^2\le 1\}$, $\Delta f=x^2y^2$, here $\Delta$ is the Laplacian. Find $$\iint_B{\left( \frac{x}{\sqrt{x^2+y^2}}\frac{\partial f}{\partial ...
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