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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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$\lim_{x \rightarrow 0} \frac{x -\sin x}{x (1 - \cos x)}$

I'm trying to find the following limit (without L'Hopital, not there yet): $$\lim_{x \rightarrow 0} \frac{x -\sin x}{x (1 - \cos x)}$$ Tanking into account that: $$\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$$ and $$\lim_{x \rightarrow 0} \frac{1 -…
Jordi
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Limit of Integral of Difference Quotients of Measurable/Bounded $f$ Being $0$ Implies $f$ is Constant

Assume $f$ is Lebesgue measurable and bounded on $[0,1]$ and that $$\int_{0}^{1}\frac{|f(x+h)-f(x)|}{h}\;dx\to0\;\text{as}\;h\to0.$$ Show that $f$ is constant for almost every $x$ in $[0,1]$. My question relates to my proposed proof below. At no…
Sargera
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Sequence of functions converging implies continuity

Let $\{f_{n}\}_{n}$ be a sequence of functions from $[0,1] \to \mathbb{R}$. Let $f:[0,1] \to \mathbb{R}$ such that for any $x\in [0,1]$ and any sequence $\{x_{n}\}$ of elements in [0,1], if $x_{n}\to x$, then $f_{n}(x_{n}) \to f(x)$. Prove that f is…
unoriginalname
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Proof of Limit Property

I am trying to show that if $f(x) \geq 0$ for every $x \in (-\infty, a)$ and $ \lim_{x \rightarrow a^-} f(x)$ exists then $\lim_{x \rightarrow a^-} f(x) \geq 0$. Even though it is intuitively obvious, the proof I have come up with is so easy it…
ItsNotObvious
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Fubini lookalike for arbitrary set

Let $A$ and $B$ be rectangles in $\mathbb{R}^k$ and $\mathbb{R}^n$ respectively. Let $S$ be a set contained in $A\times B$. For each $y\in B$, let $$S_y=\{x\mid x\in A\text{ and }(x,y)\in S\}.$$ We call $S_y$ a cross-section of $S$. Show that if…
Mika H.
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Order of differentiating in two variables

Let $A$ be open in $\mathbb{R}^2$; let $f:A\rightarrow\mathbb{R}$ be of class $C^2$. Let $Q$ be a rectangle contained in $A$. (a) Use Fubini's theorem and the fundamental theorem of calculus to show that $$\int_QD_2D_1f=\int_QD_1D_2f.$$ (b) Give a…
Mika H.
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Dense subset has at most one point on each horizontal/vertical line

Show that there is a dense subset $S$ of $A=[0,1]\times [0,1]$ such that $S$ contains at most one point on each vertical line and at most one point on each horizontal line. When I think about dense subsets and which contain "not so many" points,…
Mika H.
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Completeness of sequence of reals with finitely many nonzero terms

Let $S$ be the set of all sequences $x=\{x_n\}$ of real numbers such that only a finite number of the $x_n$ are nonzero. Define $d(x,y)=\max|x_n-y_n|$. Is the space complete? Completeness means that any Cauchy sequence converges to a point in the…
Paul S.
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Proving the real numbers are complete

In Rudin's book, the following proof is published: Let $A$ be the set of all positive rationals $p : p^2 < 2$. Let $B$ be the set of all positive rationals $p : p^2 > 2$. $A$ contains no largest number and $B$ contains no smallest. Let q be a…
Don Larynx
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Completeness and Cauchy Sequences

I came across the following problem on Cauchy Sequences: Prove that every compact metric space is complete. Suppose $X$ is a compact metric space. By definition, every sequence in $X$ has a convergent subsequence. We want to show that every Cauchy…
Damien
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If $f \in L^{2}([0, 1])$, then how do I prove that $\lim_{x\to 0} \int_{0}^{1} f(x+t)\cdot f(t)\, dt = \int_{0}^{1}f(t)^2\,dt$?

Let $f \in L^{2}([0, 1])$, and extend it to be defined on all of $\mathbb{R}$ by setting it equal to $0$ outside of $[0, 1]$. We're given the function $$F(x)=\int_{0}^{1}f(x+t)\cdot f(t)\, dt $$ and asked to prove that it is continuous at $x=0$. I'm…
Mark
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For every real function $f$, is there a strictly increasing real function $g$ above $f$?

I apologize if this question is too elementary, but I don't know how to prove the statement in my question. Suppose $f$ is a real function. Does there exist a real function $g$ such that $g \geq f$, and $g$ is strictly increasing? By $g \geq f$, I…
user107952
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Closed form solution for $1 - \left(\frac{n-x}{n+1}\right)^n = \left(\frac{n(x+1)}{n+1}\right)^n$

I was looking for a closed form solution to $$1 - \left(\frac{n-x}{n+1}\right)^n = \left(\frac{n(x+1)}{n+1}\right)^n,$$ Where we fix a $n \in \mathbb{N}$ and want to find some $x \in (-1, n).$ I know there exists exactly one solution for all $n$,…
Brayden
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$\epsilon$ definition of limit point

In the definition of a limit point $x$ of a set $A$ ($x$ not necessarily in $A$), it is required that for every $\varepsilon>0$, there exists $y\in A$ such that $$0<|x-y|<\varepsilon$$ Is this a standard definition? Do the inequalities have to be…
user90335
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Existence of square root

I am self-studying real analysis for fun and came across the following question (from A First Course in Real Analysis by Berberian). (i) Prove the following: If $r$ is a real number such that $0 \leq r \leq 1$, then there exists a unique real…
Damien
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