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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.

Real analysis is a branch of mathematical analysis, which deals with 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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Property of sup of a set of numbers

Suppose that $S\neq\emptyset$ is a bounded set of numbers and that $a$ is a number. Define $aS=\{ax\mid x\in S\}$. Prove that sup $aS$ = $a$ sup $S$ if $a \geq 0$ I can intuitively see why this is true by just trying out some cases for $a$ and $S$…
Dan H
  • 409
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Sum of non negative powers of 2

I found this very interesting. I looked at the part where it mentioned about the uppasala lectures on calculus. It mentions that in the first lecture it was proved that the sum of all the non negative powers of two equals $-1$ and even this…
Primeczar
  • 537
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Showing that $\sin(\sqrt{4 \pi^{2}n^{2} + x})$ converges uniformly on $[0,1]$

Suppose we are considering the sequence of functions $f_{n}(x)=\sin(\sqrt{4 \pi^{2}n^{2} + x})$ and I am having trouble showing that that $f_{n}$ converges uniformly on the interval $[0,1]$. An idea, I've tried is to consider the Taylor…
user135520
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Proof of $S \subset \mathbb{R}$, $ \inf(S)\leq \sup(S)$

Prove that for any nonempty set $S \subset \mathbb{R}$, $ \inf(S)\leq \sup(S)$ and give necessary and sufficient conditions for equality. This is what I have so far but I think I am on the wrong track: Since set S is contained in R, we have four…
math101
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Derivatives of sums and products "isomorphic" to powers of products and sums

Let $f$ and $g$ be smooth Real functions. From basic algebra, any power of a product is a product of powers: $$(f\cdot g)^n = f^n\cdot g^n,$$ and a power of a sum is given by the binomial expansion $$(f+g)^n = \sum_{k=0}^n \binom nk f^{n-k}g^k =…
mr_e_man
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What is wrong in my proof? (Uniform convergence and Lebesgue integral)

Rudin RCA p.32 exercise 10 Let $(X,\mathfrak{M},\mu)$ be a measure space and $\mu(X)<\infty$ and $\{f_n\}$ be a sequence of bounded complex measurable functions on $X$, and $f_n\rightarrow f$ uniformly on $X$. Prove that $\lim_{n\to\infty}\int_X f_n…
Katlus
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What am I doing wrong in my proof?

If $f$ is continuously differentiable on $[a,b]$, then show that for all $x,y\in(a,b)$ using the definition of derivative $$ |f(x)-f(y)|\leq|x-y|\sup_{x\in[a,b]}|f'(x)| $$ My attempt: Since $f'$ is continuous on $[a,b]$, it is bounded so there…
user564247
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Proving a set is bounded.

Let $E = \{x \in\mathbb R^p : \sum_{i = p}^p X_i^2/\alpha_i^2 \leq 1\}$ Prove that $E$ is closed and bounded. To prove that $E$ is closed I used the fact that the boundary of the set $E$ is equal to $\{x \in\mathbb R^p : \sum_{i = p}^p…
FSteval
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extend an alternative definition of limits to one-sided limits

The following theorem is an alternative definition of limits. In this theorem, you don't need to know the value of $\lim \limits_{x \rightarrow c} {f(x)}$ in order to prove the limit exists. Let $I \in R$ be an open interval, let $c \in I$, and let…
Lindsay Duran
  • 1,255
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Bounded nonempty countable sets necessarily contain its supremum?

Is this true or false? I know how to prove that finite sets contain its supremum. Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb R$. Since $A$ is finite, $\max(A)$ exists (can be proved by induction). And $\max(A)=\sup(A)$. So finite sets…
Yunhui Shi
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Prove that $\exists\; \delta>0$ such that $\left\lvert \frac{f(t)-f(x)}{t-x}-f'(x)\right\rvert<\epsilon$

Suppose $f'$ is continuous on $[a,b]$ and $\epsilon >0$ is given. Prove that $\exists\; \delta>0$ such that $$\left\lvert \frac{f(t)-f(x)}{t-x}-f'(x)\right\rvert<\epsilon$$ $\forall \;0<|t-x|<\delta, a\le x \le b ,a\le t\le b$. How to solve this…
Understanding math
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Computing a Limit using the Limit Definition

I've just started Real Analysis. In the textbook (Real Analysis and Applications, by Davidson and Donsig) they have defined the limit of a sequence. I am working on one of the provided exercises. No suggested solutions are provided for any…
Delightful_Richard
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Are functions which take on the values $\pm \infty$ considered unbounded?

I reading some bits and pieces of analys (some, Sakarchi and Stein, Tao). In general, if $f(0) = +\infty$ and $\forall (x \not = 0) |f(x)|<10$, would $f$ be called bounded or unbounded? Maybe more generally, is any function in the extended real…
Ovi
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proof that an alternative definition of limit is equivalent to the usual one

How to prove the following theorem? Let $I \in R$ be an open interval, let $c \in I$, and let $f: I-{c} \rightarrow R$ be a function. Then $\lim \limits_{x \rightarrow c} {f(x)}$ exists if for each $\epsilon > 0$, there is some $\delta > 0$ such…
Lindsay Duran
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Trouble understanding a part of Rudin's construction of the reals - the additive inverse of a real

I was studying Rudin's Principle of Mathematical Analysis. In his construction of the reals there is a part I am having trouble comprehending. It concerns the additive inverse of the real. I understand that for a real number $\alpha$ , which is a…
ameyask
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