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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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Can $\varepsilon$-$\delta$ definitions be used to find a limit or only to verify?

so I was wondering if there is any part of the $\varepsilon$-$\delta$ definition of the limit that offers any insight on how to find the limit of a function, or if this is something you are supposed to guess at based on the function itself or other…
user1236
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Rudin Principles Page 42: Cantor set.

No segment of the form $\left(\dfrac{3k+1}{3^{m}},\dfrac{3k+2}{3^{m}}\right)$ where $k,m\in\mathbb{Z}^{+}$ has a point in common with the Cantor set. Is there a simple proof of this statement?
MathMajor
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Mean Value Theorem to Second Derivative

Let $f$ and $g$ be twice differentiable and continuous in $[a,b]$. Show that there exists a $c$ in $[a,b]$ such that: $$\frac{f(b) - [f(a) + f'(a)(b-a)]}{g(b)-[g(a)+g'(a)(b-a)]} = \frac{f''(c)}{g''(c)}$$ I was thinking by MVT applied to both $f$ and…
Aaron
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Ratio test implies Raabe's test

I want to prove the ratio test $$\lim \sup \left| \frac{a_{n+1}}{a_n} \right|<1$$ implies the Raabe's test $$\lim \sup \left| \frac{a_{n}}{a_{n+1}} \right|>1 + \frac{C}{n}$$ for $C>1$ I am doing the following: $$\lim \sup \left( \left|…
user186918
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Convergence of single variable power series

I would like to know whether the following power series converges or diverges. $$1-x+\frac{x^2}{2!}-\frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$ My intutition tells me that for any nonzero $x$, the series diverges, but I am not sure how to verify…
mononono
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Bolzano-Weierstrass application?

I am having problems proving the following claim: Given a bounded set $A \subset R^n$, I want to prove the existence of $a_1, \dots, a_N \in R^n$ and numbers $r_1, \dots, r_N \in [0, +\infty)$ such that $$A \subset \cup_{k=1}^N B(a_k,r_k)$$ Where…
user186918
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Continuous extension of a real function defined on an open interval

Let $I\subset\mathbb{R}$ be a compact interval and let $J$ denote its interior. Consider $f:J\to\mathbb{R}$ being continuous. Under which conditions does the following statement hold? $$ \text{There exists a continuous extension $g:I\to\mathbb{R}$…
precarious
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How do I prove this form of mean value theorem for integral?

I only knew the standard mean value theorem for integrals. (i.e. $\int_a^b f(x)dx= f(c)(b-a)$ for some $c$ between $[a,b]$ where $f$ is continuous. This is directly derived by applying mean value theorem and Fundamental theorem of calculus) I'm…
Rubertos
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Limit Theorems Without Magic Epsilons

Do there exist proofs of the major limit theorems for sequences that don't involve picking magic epsilons? I call such epsilons magic because they seem to appear in the proof out of thin air and indeed they work, but where they come from is unknown.…
ItsNotObvious
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derivative bounded by a constant multiple of the function

$f$ is differentiable on $[a,b]$, $f'(x) \leq A|f(x)|$ where $A$ is a non-negative constant. If $f(a)=0$ show $f(x)=0, \forall x\in [a,b]$ I imagine the proof uses the Mean Value Theorem but I have not been able to get it to work. I know…
user9352
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Supremum of a set of irrational numbers

I need help with the following example: Let S be the set of all irrationals in $[0,1]$. Show that $\sup(S) = 1$. Is there some property that I should be referring to when proving problems like these? The set definition states that it is bounded…
Pandamonium
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Proof of differentiability implies continuity in higher dimenstion.

There is theorem that says if a function is differentiable at a point in an open set, then it is continuous at the point. This can be proven by doing : $$\lim_{x \rightarrow a} f(x)-f(a) = \lim_{x \rightarrow a} (x-a) \left(…
eChung00
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Equivalence class on real numbers

Call two real numbers equivalent if their binary decimal expansion differ in a finite amount of places, if S is a set which contains an element of every equivalence class, must S contain an interval? How to show that every interval contains an…
TROLLHUNTER
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Regarding the Poincare Friedrichs inequality

I am working on a two part problem. Part 1 was to prove the Poincare-Friedrichs inequality for n=1: $\int_{0}^{\alpha} |f(t)|^2 dt \le C\int_{0}^{\alpha} |f'(t)|^2 dt$ for some constant $C$. I managed to do this using the Cauchy-Schwarz inequality.…
fideo
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Prove [(xy=0)∧(x,y∈ℤ)]→[(x=0)∨(y=0)]

I'm working through a higher algebra textbook. It has some exercises related to the positive integers and I'm stuck on this proof. Here's what I have so far: Attempted proof Assume the contrary,…
nathan.j.mcdougall
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