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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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Prove that $\exp(x)>0$ using only formal definition of exp

This problem would be easy if I could use the fact that $\exp(x)=e^x$, but I have to use the following definition: $$\exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$ I can also use the fact that $$\exp(x+y)=\exp(x)\exp(y)$$ So how do I prove, using those…
Dunno
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Measure zero and compact then content zero

I'm trying to show that a compact set $E$ with measure zero has content zero. It seems simple because for every $\varepsilon$ I take a subcover of the rectangles, but the issue I'm facing here is that I can't take a subcover because the rectangles…
user112547
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Continuous, bijective function from $f:[0,1)\to \mathbb{R}.$

Prove that there does not exist a continuous, bijective function $f:[0,1)\to \mathbb{R}.$ By contradiction I can assume a function exists, so that function is surjective, onto and continuous. And I know I need to use the intermediate value theorem…
user104235
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Is it possible that all subseries converge to irrationals?

Does there exists a positive decreasing sequence $\{a_i\}$ with $\sum_{i\in\mathbb{N}} a_i$ convergent, such that $\forall I\subset\mathbb{N},\sum_{i\in I}a_i$ is an irrational number? Such an example would give rise to a closed perfect set…
Anonymous Coward
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The measure of the diagonal of a unit square in an alternative measure.

Usually, we say that the measure of the diagonal of a unit square is 0, but that's with the preassumption the measure is Lebesgue measure in $\mathbb{R}^2$. But what if we are talking about a strange measure where its the product of a 1-dimensional…
xzhu
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Showing a recursive sequence is Cauchy

let $a_1=1$, $a_2=2$, and define $a_n=\frac{1}{2}(a_{n-1}+a_{n-2})$. How can I show that this sequence is Cauchy? I began with $|a_n-a_{n-1}|=\frac{1}{2}(a_n-a_{n-1})$ which goes to $\frac{1}{2}(a_{n-1}+a_{n-2})-\frac{1}{2}(a_{n-2}+a_{n-3})$ I am…
Kyle H.
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Prove that $\sqrt{2}$ is a real number.

I remember I saw this question somewhere in Lang's undergraduate real analysis. Given any real number $\ge0$, show that it has a square root.
user5402
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Is it possible to have a convergent subsequence of a divergent sequence?

Is it possible to have a convergent subsequence of a divergent sequence? Thanks!
eChung00
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Does the uniform continuity of $f: X \rightarrow \mathbb{R}$ imply $f: A \rightarrow \mathbb{R}$ is also uniformly continuous, when $A \subset X$?

I've been preparing for the prelim in August, and was working on a problem involving uniform continuity and restriction of functions. I absentmindedly assumed the above by considering the contrapositive: if $f: A \rightarrow \mathbb{R}$ isn't…
JakeR
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Approximation of Riemann integrable function with a continuous function

I have proved that if $f \in R[a,b]$ and given $\epsilon > 0$ there exists a continuous function $g$ such that $$\int_a^b |f-g| < \epsilon$$ I was wondering if using this fact there is some way to show that there is also some continuous function $h$…
alejopelaez
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$f'(x)=f(x)$ and $f(0)=0$ implies that $f(x)=0$ formal proof

How can I prove that if a function is such that $f'(x)=f(x)$ and also $f(0)=0$ then $f(x)=0$ for every $x$. I have an idea but it's too long, I want to know if there is a simple way to do it. Thanks! Obviously in a formal way.
Trafalgar Law
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Number of real roots of $\sum_{k=0}^{n}\frac{x^{k}}{k!}=0$

Prove the following, without induction. Is it possible? The equation $\sum_{k=0}^{n}\frac{x^{k}}{k!}=0$ has no real root if $n$ is even. And if $n$ is odd, it has only one real root. I also tried searching the proofs many times with…
chloe_shi
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$ \lim\limits_{n \to \infty} \frac1n\sqrt[n]{n\cdot(n+1)\cdots(2n)}$

It tried to solve this limit $$ \lim_{n \to \infty} \frac{\sqrt[n]{n\cdot(n+1)\cdots(2n)}}{n}$$ $ \frac{\sqrt[n]{n\cdot(n+1)\cdots(2n)}}{n} = \sqrt[n]{\frac {2n!n}{n!}} \frac{1}{n} \sim \sqrt[n]{\frac { \sqrt {2 \pi 2 n} (\frac {2n}{e})^…
Anne
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if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$

Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous on $\mathbb{R}$ and that $f(r)=0$ for all $r \in \mathbb{Q}$. Prove that $f(x)=0$ for all $x \in \mathbb{R}$. My attempt: Define a sequence $(x_n)$ where $x_n \in \mathbb{Q}$ for all…
Idonknow
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The set of Lipschitz continuous bounded functions is dense in the set of bounded uniformly continuous functions.

Let $(X,d)$ be a metric space. Let $f$: $X$ $\rightarrow$ $\mathbb{R}$ be uniformly continuous and bounded. The method of attack is to consider the sequence of functions {${f_n}$} defined by $$ f_n(x) = \inf_{y\in X}\{f(y)+n\text{d}(x,y)\}.$$ It's…
beforepim
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