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.

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Convergence of vectors in $\mathbb{R}^m$ proof - help understanding a step?

The theorem is as follows: A sequence of vectors in $\mathbb{R}^m$ ($m \in \Bbb Z_+$) converges iff all $m$ sequences of its components converge in $\mathbb{R}^1$. I can follow the (if) direction of ...
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Convergence of Bisection, Secant and Newton's method when there is no root

If there is no root to a function can the bisection, secant or Newton's method still converge to some number e.g. $f(x) = \frac{1}{x}$ on interval $[-1,1]$?
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inverse function theorem: $f$ is invertible(smooth inverse), then Jacobian determinant is not zero?

Source: "An Introduction to Manifolds" by Loring W. Tu, p339, p68 For a proof, see Rudin "Principles of Mathematical Analysis" P221 But, if the map $f$ is invertible(has a $C^\infty$ inverse) in ...
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Showing every continuous function $f$ on $[a,b]$ is the uniform limit of a seq of polynomials $(q_{n})$, where $q_{n}(x) = x^{n}p_{n}(x)$

If $0 \notin [a,b]$, show that every continuous function $f$ on $[a,b]$ is the uniform limit of a seq of polynomials $(q_{n})$, where $q_{n}(x) = x^{n}p_{n}(x)$ for polynomials $p_{n}$. There was a ...
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How can I prove this :$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$ for high school level?

I have tried to evaluate this limit: $$\lim_{x\to \infty } \left(x(x+1) \log \left(\dfrac{x+1}{x} \right)-x\right)=\frac12$$ using $\lim_ {x\to \infty }\left(1+\dfrac{1}{x}\right)^{x}=e$, and using ...
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Doubt regarding greatest lower bound property implies least upper bound property

Least Upper Bound property - A set $X$ is said to have this property if every non empty subset $A$ of $X$ which is bounded above has the least upper bound. (This does not imply the least upper bound ...
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What is the mathematical meaning of the following sentence

What is the mathematical meaning of the following sentence: The function $f(x)$ is non-decreasing as $x$ decreases to $0$.
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For what continuous $f$ we can write $f(x)+a\cdot f(x+\alpha)$ as $b\cdot f(x+\beta)$?

After reading about harmonic addition theorem, I am interested in: Find all continuous $f:\mathbb R\mapsto\mathbb R$ that admit identities of the form $$f(x)+a\cdot f(x+\alpha)=b\cdot f(x+\beta)$$ ...
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Can we deduce that this sequence converges to zero

Let $(x_{k})_{k≥1}$ be a real sequence verifying $x_{k}=0$ infinitely many times. When $x_{k}≠0$, then $x_{k}>0$ and the values of $x_{k}$ decreases to zero when $k$ increases. My question: Can we ...
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Norm $||.||$ on $C(X)$ is equivalent to $||.||_{\infty}$ norm if evaluation linear functionals on $(C(X),||.||)$ is continuous.

Let $X$ be compact Hausdorff space. Let $||.||$ be a norm on $C(X)$ which makes it into a Banach space and also assume that the linear functional $\lambda_x$ given by $\lambda_x(f)=f(x)$ is continuous ...
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Uniform convergence of the series $\sum_{n=1}^\infty \frac{1}{n} (\frac{x-1}{x+1})^n$

I'm studying the series $$\sum_{n=1}^\infty \frac{1}{n} \cdot (\frac{x-1}{x+1})^n$$ I have already shown that there is pointwise convergence for $x\in[0, +\infty)$ and total convergence in any ...
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Is it possible to utilize the convergence of the sequence $z_{n+1}=a/(1+z_n)$ to prove that the sequence $x_{n+2} = \sqrt{x_{n+1} x_n}$ is convergent?

I'm doing Problem II.4.6 in textbook Analysis I by Amann/Escher. For $x_0,x_1 \in \mathbb R^+$, the sequence $(x_n)_{n \in \mathbb N}$ defined recursively by $x_{n+2} = \sqrt{x_{n+1} x_n}$ is ...
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Does $\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}\to \int_0^{1-}\frac{1}{\sqrt{1-x^2}}$?

Consider the sum $$\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}.$$ Obviously, $$\sum_{k=0}^{n-1}\frac{1}{\sqrt{n^2-k^2}}=\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{\sqrt{1-\frac{k^2}{n^2}}}.$$ I would like to ...
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A generalized differential equation for a convolved differential operator.

The solution to perhaps the world's very first differential equation $$f'(x) = f(x)$$ is the well known exponential functions $$f(x) = k\exp\left[x\right]$$ But what if we consider another ...
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is operatory norm weak* - lower semicontinuous?

Let $X$ be a normed space with norm $\|.\|$. We already know that the norm $\| . \|: X \to R$ is weakly lower semicontinuous in the sense, if $x_n \overset{w}{\longrightarrow} x_0$ as $n \to \infty$, ...
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Space of Trigonometric polynomial of degree at most n form subalgebra

I had encountered following in Real analysis by N L Carothers Page 170-172 From this $T_1$ becomes subalgebra . but if we consider $\sin x\in T_1$ , $\sin x.\sin x=\sin^2 x\notin T_1$ so how it ...
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Show uniformly convergent sequence of contraction maps has a fixed point but is not unique?

This question may have appeared in the community but has not been shown yet. Let $(M,d)$ be a compact metric space and for each $n$ in $\mathbb{N}$ let $f_n$ be a contraction mapping. Suppose $(f_n)$ ...
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Intersection of open sets of infinite measure [on hold]

Does the intersection open sets of infinite measure converge to a given set? That is, does $$\bigcap_{k=1}^{\infty} \;\; (k,\infty)$$ converge to a set? Edit: Not asking what the limit of the ...
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The fourier series of periodic and real analytic function

Let $f$ be a real analytic and periodic function defined on the interval $[0, 2\pi]$. Then $f$ is infinitely differentiable for sure. Therefore, the fourier coefficients of $f$ decay faster than any ...
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Formulas for pointwise and uniform convergence

I am looking at formulas for testing pointwise versus uniform convergence from Kenneth Ross' text, which are respectively: $$\lim _{n \rightarrow \infty} f_n (x) = f(x), \ \text{for all } x \in S$$ ...
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Showing inverse of multivariate holomorphic map is holomorphic using real analysis techniques

I'm grading for a graduate real analysis course and looking through the text, trying to do some problems before the term starts so I can stay ahead of things. I found the following exercise in the ...
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Integral of positive Part

Is $$\int_a^b \big(f(x)\big)^+\mathrm{d}x = \left( \int_a^b f(x) \mathrm{d}x \right)^+$$ provided that $f:[a,b]\to\mathbb{R}$ is integrable? This means, can taking positive part and integration be ...
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Prove that $f(1)-f(0)=\int_0^1 f'(t)dt$ using $f(x+h)=f(x)+f'(x)h+o(h)$.

Let $f:[0,1]\to \mathbb R$ a $\mathcal C^1([0,1])$ function. I want to prove that $$f(1)-f(0)=\int_0^1 f'(t)dt$$ using $f(x+h)=f(x)+f'(x)h+o(h)$. In the official solution of my lecture they do as ...
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Proof Correction: if $f = f'$, then $f = e^x$

My textbook states that if $f = f'$, then $f = ce^x$. I can't see the flaw in my proof however (which is totally different from the textbook's). Proof $f$ keeps the same sign. Suppose otherwise, $f$ ...
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Show existence of a solution in differential equation for analytic function

Question: Suppose we have a differential equation $$y'(x) = \phi (y(x))~,$$ where $~\phi~$ is analytic, i.e. $$\phi = \sum_{k=1}^{\infty} a_k x^k\qquad \text{with~~\qquad limsup} |a_k|^{1/k} = 0~.$$...
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