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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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Transfinite derivatives

I don't know if this is exactly research level, as I am only starting college. But I feel like this is the best place to ask the question. We all know of 1st, 2nd, 3rd, nth derivatives. Is there a way of extending it to the transfinite ordinals?…
user107952
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Proving that $(b_n) \to b$ implies $\left(\frac{1}{b_n}\right) \to \frac{1}{b}$

In my textbook (S. Abbott. Understanding Analysis 1 ed. pp 47 Theorem 2.3.3.iv), the author proves $b_n \to b$ implies $\frac{1}{b_n} \to \frac{1}{b}$ the following way: $$\left|\frac{1}{b_n}-\frac{1}{b}\right|=\frac{|b-b_n|}{|b||b_n|}$$ therefore,…
confused
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If two continuous functions are equal almost everywhere on $[a,b]$, then they are equal everywhere on $[a,b]$

Suppose $f$ and $g$ are continuous functions on $[a,b]$. Show that if $f=g$ almost everywhere on $[a,b]$, then, in fact, $f=g$ on $[a,b]$. Is a similar assertion true if $[a,b]$ is replaced by a general measurable set $E$? I have known that the set…
Yang
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Baby Rudin Theorem 1.20 (b) Proof

I have a question about Rudin's proof of Theorem 1.20 (b) in his book Principles of Mathematical Analysis. Theorem 1.20 is stated as follows: (a) If $x\in R, y\in R$, and $x>0$, then there is a positive integer $n$ such that $$nx>y.$$ (b) If…
Zilliput
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Clarification of L'Hopital Proof Pugh

I am self-studying Real Analysis right now via Pugh's Real Mathematical Analysis but am having trouble understanding a step of the author's proof of L'Hopital's rule. The theorem is stated as: If $f$ and $g$ are differentiable functions defined on…
Lewis Wagner
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3 answers

Let $(x_n)$ be a bounded but not convergent sequence. Prove that $(x_n)$ has two subsequences converging to different limits.

Let $(x_n)$ be a bounded but not convergent sequence. Prove that $(x_n)$ has two subsequences converging to different limits. My attempt is: Since the sequence is bounded , there exists $M>0$ such that $x_n \in [-M,M]$ for all $n \in…
Idonknow
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Is every subset of a metric space a metric subspace?

Is every subset of a metric space a metric subspace? A simple proof does justify that all are subspaces, still, wanted to know if I missed something.
Jesse P Francis
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Prove that there exists $f,g : \mathbb{R}$ to $\mathbb{R}$ such that $f(g(x))$ is strictly increasing and $g(f(x))$ is strictly decreasing.

Prove that there exists $f,g : \mathbb{R}$ to $\mathbb{R}$ such that $f(g(x))$ is strictly increasing and $g(f(x))$ is strictly decreasing. I tried cases by taking $f(x)$ as an increasing function and $g(x)$ as a decreasing function then I am…
user321656
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4 answers

Uniform convergence of geometric series

How do I show that the geometric series $\sum_{k=0}^\infty x^k$ converges uniformly on any interval $[a,b]$ for $-1 < a < b < 1$? The Cauchy test says that $\sum_{k=0}^\infty x^k$ converges uniformly if, for every $\varepsilon>0$, there exists a…
Orange
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Difference between limsup and sup

I'm having some difficulty visualising the difference between the limit supremum and supremum (and for limit infimum/infimum) for bounded sequences. Would it be possible for some to provide a brief explanation and maybe some examples?
Inazuma
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Showing that the space $C[0,1]$ with the $L_1$ norm is incomplete

Can anyone think of a relatively easy counter example to remember, which demonstrates that the space $C[0,1]$ with the $L_1$ norm is incomplete? Thanks!
user26069
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3 answers

Uniform convergence of sequence of convex functions

Let $\{f_n\}$ converge point-wise to $f$, where each $f_n:[a,b]\rightarrow \mathbb{R}$, and each $f_n$ is a continuous convex function. Furthermore, assume that $f$ is continuous. Prove that the convergence is uniform. I was trying to do something…
Daniel Montealegre
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Continuity and Joint Continuity

Consider a function $f(x,y):[0,1] \times [0,1] \rightarrow R.$ What is the difference between $f$ continuous in each argument and jointly continuous?
user12586
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Show that a set is dense in $[-1,1]$

Show that $\{\cos\ n:n \in \mathbb{N}\}$ is dense in $[-1,1]$ by using the fact below: Suppose $x$ is irrational. Then there exists $p_n,q_n \in \mathbb{Z}$ such that $\bigg|x -\frac{p_n}{q_n}\bigg| < \frac{1}{q_n^2}$ I have no idea on how to…
Idonknow
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4 answers

If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous

From Pugh's analysis book, prelim problem 57 from Chapter 4: Let $f$ and $f_n$ be functions from $\Bbb R$ to $\Bbb R$. Assume that $f_n(x_n)\to f(x)$ as $n\to\infty$ whenever $x_n\to x$. Prove that $f$ is continuous. (Note: the functions $f_n$ are…
Aden Dong
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