# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

93,718 questions
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### Proof of Density of Rational Numbers as given in Howie

I'm working out of John M. Howie's Real Analysis (2001). In section 1.4 (Exercise 1.5), we're asked the following: Let $x$ and $y$ be real numbers, with $x < y$. Show that, if $x$ and $y$ are ...
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### Suppose $x_n$ is a decreasing sequence of positive reals with $\sum x_n$ converges, must $(n\log n)x_n \to 0$

We are able to show, using the Cauchy criterion (using sum from $n$ to $2n$) that $nx_n \to 0$ Explicitly this is $0<nx_{2n}<\displaystyle \sum_{i=n}^{2n}x_n$ and the result follows from squeeze ...
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### When is Taylor series a polynomial?

We know that Taylor series is a rational function if and only if its coefficients satisfy the linear recurrence relation. Can we put further conditions on the coefficients so that the Taylor series ...
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### ‎If ‎$‎\lim‎ f(x) = L‎$ as ‎$‎x‎\rightarrow ‎+‎\infty‎$‎, then $‎\lim‎ f(c x) = L‎$ as $‎x‎\rightarrow ‎+‎\infty‎$.

‎Let ‎‎$‎f‎$ ‎be a‎ ‎real-variable ‎function ‎such ‎that ‎‎$‎‎‎‎‎\lim‎‎ f(x) = L‎$ as ‎$‎x‎‎\rightarrow ‎+‎\infty‎$ where $‎L\in‎\mathbb{R}‎$‎. ‎Also, let‎ $‎c‎>0‎$ be a constant. My question is:‎ ‎...
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### About Baire's theorem

Exercise $3.22$ of baby Rudin is to prove the Baire's theorem: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, ...
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### Prove that $f: X \rightarrow R$ is continuous with respect to the metric $d_1$ on $X$ iff it is continuous with respect to the metric $d_2$ on $X$.

Let $X$ be a set and $d_1, d_2$ be metrics on $X$ so that for constants $m,M > 0$ and any $x,y \in X$ we have $md_1(x,y) \leq d_2(x,y) \leq Md_1(x,y)$ Prove that $f: X \rightarrow \mathbb{R}$ is ...
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### A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
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### Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
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### Uniformly Bounded and Bounded Variation [on hold]

Studying functions of bounded variation, the following exercise came up: Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of functions with $f_n:I \to \mathbb{R}$. Show that: If $(f_n)_{n\in \mathbb{N}}$...
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### Proof verification: At most countably many local maxima

I'd appreciate a second pair of eyes on a proof. I want to prove that a function $f:\mathbb{R}\to\mathbb{R}$ can have at most countably many strict local maxima. The question has been asked elsewhere ...
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### Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
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### Let $f:[a,b] \rightarrow \mathbb{R}$ is cts with $f(a), f(b) < 0, \int_a^b f(x)dx \ge 0$. Show $\exists c \in (a,b) s.t. \int_c^b f(t) dt \ge 0$.

My solution to this problem is the following: Solution: $[a,b]$ is closed interval, so $f$ is uniformly continuous on the interval $[a,b]$. So, we know that there exists $\delta >0$ such that ...
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### Set construction confusion

Let $S$ be an infinite bounded subset of $\mathbb R$. Now let's construct a set $T$ where, $$T=\{x:x \textrm{ exceeds at most a finite number of elements of } S\}.$$ Is there any element of $T$ which ...
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### Let $(X,d)$ be a metric space and $A\subseteq X$ be compact. Prove that for any $y \in X$ there exists $x \in A$ so that $d(y,x) = d(y,A)$ [duplicate]

Let $(X,d)$ be a metric space and $A \subseteq X$ be compact. Prove that for any $y \in X$ there exists $x \in A$ so that $d(y,x) = d(y,A)$ where $d(x,y)=|x-y|$. Since A is compact it is ...
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### Find the closure and the interior of A

I have to find the closure and the interior of the set A defined by $A =${$(x,\sin(x^{-1})) : x \in R-\{0\}$} $\subset$ $R^2$ I don't know how to start. I know that $\sin(x^{-1})$ has it's maximum ...
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### Is this product strictly positive? [on hold]

Let $p\in (0,1)$ and $\varepsilon\in (0, 1)$ be fixed. For all $i\in \mathbb N$ we define $p_i=1+(p-1)(1-\varepsilon)^i$. Is it possible to prove that $$\prod_{i=1}^{+\infty}\frac{p_i}{2-p_i}>0$$? ...
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### What does it mean for something to be strictly less than $\epsilon$ for an arbitrary $\epsilon$?

Perhaps a trivial question, but something I never completely understood. If we have shown that $a-b < \epsilon$ for all $\epsilon > 0$, then does that imply that $a-b \le 0$? I"m interested in ...
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### Radius of convergence comparison for power series

Given two power series $g (x)=\sum a_n x^n$ and $h (x)=\sum b_n x^n$ with radius of convergence $R_1$ and $R_2$ such that $g(x) \leq h (x)$ for all $x \in \{|x| \leq min \{R_1,R_2\}\}$, Does this ...
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### The behaviour of functions nested under themselves outwith their domains

This question follows from some interesting observations on a sum of reciprocals. Instead of summing them however, we will place each fraction to make a continued fraction. Some visualisations on ...