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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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How to prove a convergence result?

I met a problem like following: If $E\subseteq \mathbb{R}$ is Lebesgue measurable and its measure is finite. Show that $$\lim_{n\to\infty}\int_Ee^{inx}\,\mathrm{d}x=0.$$ A brutal attempt of using the Dominated Convergence Theorem fails as we do not…
OnoL
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Let $f(x)$ be a continuous function in $\mathbb{R}$, and let $(a_n)$ be a Cauchy sequence. Prove that $f(a_n)$ is a Cauchy sequence.

This is a question I've stumbled upon and the question asks me to prove something which I don't think is true. Question: Let $f(x)$ be a continuous function in $\mathbb{R}$, and let $(a_n)$ be a Cauchy sequence. Prove that $f(a_n)$ is a Cauchy…
sillyme
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Prove that the limit of a sequence in a closed set $F$ is in $F$

Say $F$ is a closed subset of $\mathbb{R}$ and let ${p_n}$ be a sequence in $F$ converging to $p \in \mathbb{R}$. We want to prove $p \in F$. I think I did some point wrong but can not figure out: If ${p_n}$ converges then there exist $\epsilon$ >0,…
Kingkong
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h(xy) = h(x)h(y) Proof

Let $h$ be a differentiable function where $h(xy) = h(x)h(y), \forall x>0$. Let $h(1)=1$ be an initial condition. Prove $\exists c$ such that $h(x)=x^c$. I've tried differentiating both sides of the original relation, but I seem to be doing…
Quentin Tarantino
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Examples of flat function

A flat function is a smooth function $ƒ : \mathbb{R} → \mathbb{R}$ all of whose derivatives vanish at a given point $x_0 \in \mathbb{R}$. Can anybody suggest a non-trivial example of function flat at $x_0=0$ which is not an "obvious" variant of…
user95731
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What to do with $xe^x$?

How can I solve this for $x$? $xe^x=-2/a$ with $(a \in \mathbb{R_0^+})$ $a$ can be any strict positive real number. I need this because I'm searching for the root of a function to sketch a graph.
Mats
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Real Analysis: Prove that the function $f$ is uniformly continuous on the interval $[0, \infty)$.

Prove that the function $$f(x) = \frac1{1+x^2}$$ is uniformly continuous on the interval $[0, \infty)$. One part of my proof is $$\begin{align*} \left|\frac1{1+x^2}-\frac1{1+y^2}\right| &= \left|\frac{y^2-x^2}{(1+x^2)(1+y^2)}\right| \\ & =…
afsdf dfsaf
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Inequality $\frac{x}{1+x} < \ln(1+x), \forall x>0$

Prove that $\frac{x}{1+x} < \ln(1+x), \forall x>0$. I wrote it as $e^x < 1 + x + (1+x)^x$ to see if it would make it any simpler. I do not think induction would work since that only works for natural numbers (right?). I also tried writing it as $1 +…
AnthonyT
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Bounded variation function on real line as difference of increasing functions

A bounded variation (BV) function $f$ on the interval $[a,b]$ can be written as a difference of two monotone increasing function. This can be done by a construction where $$F(x):=\sup \sum_{j=1}^{n-1}|f(x_{j+1})-f(x_j)|$$ where the supremum is taken…
JJ Beck
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Definition of logarithm function derived from its useful properties

My understanding of the point of logarithms is that they turn multiplication into addition, and exponentiation into multiplication. i.e. $$ \ln cx = \ln c + \ln x $$ $$ \ln x^c = c \ln x $$ Let's call the above two statements about logarithms their…
Paul Hollingsworth
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Eliminating discontinuities for increasing function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a monotone increasing function. We know from a general fact that the set of discontinuities of $f$ is countable. Denote the set of discontinuities by $D$. Define $g:\mathbb{R}\rightarrow\mathbb{R}$ as…
PJ Miller
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Bounded variation function is countinuous except at countably many points

A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a BV function if there exists $M<\infty$ for which $$\sum_{k=1}^N|f(x_k)-f(x_{k-1})|\leq M$$ for every sequence $x_0
JJ Beck
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What's $F'(x)$ if $F(x) = \int_a^{g(x)} H(x,t) dt$?

I was using the chain rule and i have $F'(x) = H(x, g(x)) \cdot g'(x)$. Is this right?
Mario De León
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Prove that $\left | \cos x - \cos y \right | \leq \left | x - y \right |, \forall x,y \in \mathbb{R}.$

This is one of the problem in my text book where the section which the problem is stated talks about the mean value theorem and Rolle's theorem. By looking at this, I have no idea where to start. Can I have some hints??
eChung00
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Riemann integral property proof

Completely lost. I know I have to use the sup and it is an epsilon delta proof, but other than that, I am confused.
Skuttle_Butt
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