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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. .

145439 questions
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$f'$ has maximal rank

I'm trying to prove that the set of orthogonal matrices $O(n)$ is a manifold. (We identify the set of matrices of order $n$ with $\Bbb R^{n^2}$) In order to do so, I must show $f'$ has maximal rank in $O(n)$, having already defined $f:\Bbb…
user108811
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Restriction of vector field tangent to sphere

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. Show that the vector field $$w=x_1\frac\partial{\partial…
PJ Miller
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Taylor's theorem: $f'' + f = 0, f(0) = f'(0) = 0$.

I am having a hard time coming up with a solution to this problem. Suppose that $f$ is twice differentiable and that $f'' + f = 0$. If $f(0) = f'(0) = 0$, use Taylor's theorem to show that $f = 0$. The definition of Taylor's theorem we were given…
louiswins
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The Rapidity of the Exponential Function Towards Infinity

Since $$ \begin{align*} & e^x = 1 + x + \ldots + \frac{x^n}{n!} + \frac{x^{n+1}}{(n+1)!} + \ldots , \\ & x^{-n}e^x \gt \frac{x}{(n+1)!} \rightarrow \infty \end{align*}$$ when $ x \rightarrow \infty $. Hence $ e^x $ tends to infinity more rapidly…
Hemofelicity
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Prove that if $A$ is compact and $B$ is closed and $A\cap B = \emptyset$ then $\text{dist}(A,B) > 0$

Let $X$ be a metric space. For nonempty subsets $A,B\subseteq X$. Define $\text{dist}(A,B) := \inf\{d(x,y) : x\in A, y\in B\}$ a) Prove that if $A$ is compact and $B$ is closed and $A\cap B = \emptyset$ then $\text{dist}(A,B) > 0$ b) Suppose that $X…
JohanLiebert
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Suppose $\{x_n\}$ is a monotonically increasing sequence and the subsequence $\{x_{2n}\}$ is convergent. Justify $\{x_n\}$ is convergent.

I think I did this correctly: Since $\{x_{2n}\}$ is convergent, it is bounded. Since $\{x_{2n}\}$ is a bounded subsequence of $\{x_{n}\}$, $\{x_{n}\}$ is also bounded. Hence, $\{x_{n}\}$ is monotone and bounded. Therefore, $\{x_{n}\}$ is convergent…
Simon
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Image of connected set is connected implies continuity?

Let $f\colon \mathbb{R} \mapsto \mathbb{R}$ be a function with the property that the image of every connected set is connected. Is $f$ necessarily continuous? I've recently learned the definition of connected set and i'm still not totally…
u1571372
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Special case of Riemann Rearrangement Theorem

Let $\sum_{k=1}^\infty a_{\varphi(k)}$ be a rearrangement of a conditionally convergent series $\sum_{k=1}^\infty a_k$. Prove that if $\{\varphi(k)-k\}$ is a bounded sequence, then $\sum_{k=1}^\infty a_{\varphi(k)}=\sum_{k=1}^\infty a_k$. I can't…
user98964
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$f''$ has zero on interval $(a,b)$ if graph of $f$ intersects line between $(a,f(a))$ and $(b,f(b))$

Let $ f$ be continuous on $[a,b] $ and assume the second derivative $f''$ exists on (a,b). The graph of $f$ and the line segment joining the points $(a,f(a))$ and $(b,f(b))$ intersect at a point $(x_{0},f(x_{0}))$ where $a
Redsbefall
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Partition of unity on neighborhood of compact set

Let $U\subseteq\mathbb{R}^n$ be open and $D\subseteq U$ be compact. Prove that there is a $C^{\infty}$ function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $f$ takes on the value $1$ on a neighborhood of $D$, and the support of $f$ is contained…
Paul S.
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Every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ with $ a_k≤ k − 1$ for $k ≥ 2$ .

I have this theorem which I can't prove.Please help. "Show that every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ in one and only one way where each $a_k$ is non-negative integer with $ a_k≤ k − 1$ for $k…
Silent
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Prove that if two open balls in a vector space are equal then their radii and centres are equal

Define open balls in a metric space $(X,d)$ with center $a$ and radius $r$: $B_r(a) = \{x\in X: d(x,a) < r\}$ Let $(V, \|*\|)$ be a normed vector space, (with the corresponding metric $d(x,y) := \|x - y\|)$. Let $a,b \in V$ and let $r,s > 0$.…
JohanLiebert
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5 answers

Constructing a set with exactly three limit points

Construct a set of real numbers having exactly three limit points. Progress: I know the set $\{ 1/n + k\}$ has one limit point $k$, but I am unable to justify why. Please be a bit elaborate in your explanation because I need to understand this…
Charlie
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Taxicab metric with open, closed unit ball, and unit sphere.

In $\mathbb{R}^2$, the Taxicab metric is defined by $d(x,y) = |x_1 - y_1| + |x_2 - y_2|.$ In this metric, describe/draw the open unit ball, closed unit ball, and the unit sphere which are based at the origin with radius $1$. Here is my work: Since…
therexists
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Cluster Point Theorem

I came across the following problem about cluster points: Prove the following: $K$ is a cluster point $\Longleftrightarrow$ $K$ is the limit of some subsequence $\{a_{n_i}\}$. This is my attempt: Proof. $(\Leftarrow)$: Suppose $K$ is the limit of…
Damien
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