Articles of real analysis

Proof Check: Every Cauchy Sequence is Bounded

Sorry if I keep asking for proof checks. I’ll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I’d do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I’m not sure if […]

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 4: How to show boundedness?

Let $f_1$, $f_2$ be the functionals defined on the normed space $C[a,b]$ of all continuous functions defined on the closed interval $[a,b]$ with the maximum norm be defined as follows: $$f_1(x) \colon= \max_{t\in[a,b]} x(t), \; \; \; f_2(x) \colon= \min_{t\in[a,b]} x(t) \; \; \; \forall x \in C[a,b].$$ Then $f_1$ and $f_2$ are not linear […]

On epsilon delta definition fo Limit.

Is it possible to use $\delta=\min(1,\frac\varepsilon c)$ in the following exercise? Thanks in advance!! $$\lim_{x\to 1}(x^2+4x)=5.$$ To make my question clear, do we have a right to choose the limit constant a as $\delta$?

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Fix some positive integers $L$ and $k \leq L$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $L – \sum\limits^N_{i=1} r_i < k$ and all the $r_i \geq k$. Prove (or disprove) that the minimum of $$ \mathcal{H}(r) […]

Prove Intersection of Two compact sets is compact using open cover?

Let A and B be compact subset of R To show intersection of A and B is compact, I need to show that for any open cover for intersection has finite subcover. It is quite straightforward for Union of two compact sets, but how can I start with the intersection casE?

In search of easier manipulation of an inequality to prove it

Let the zigzag function defined as $$zz(x):=\left|\lfloor x+1/2\rfloor-x\right|,\quad x\in\Bbb R$$ Then for $k\in\Bbb Z$ we have that $zz(k)=0$, and $zz(\Bbb R)=[0,1/2]$, and $zz$ is increasing in any interval of the kind $[k,k+1/2]$, and decreasing in $[k+1/2,k+1]$ for $k\in\Bbb Z$. Then we define the function $$F(x):=\sum_{n=0}^\infty \frac{zz(4^nx)}{4^n},\quad x\in\Bbb R$$ Then I need to prove that $F(x)$ […]

Prove that $C^1_0$ with a certain metric is a complete metric space.

$C^1_0[0,1]$ is the set of functions $f:[0,1]\rightarrow \Bbb R$ such that $f,f’$ are continuous on $[0,1]$ and $f(0)=0.$ The metric $d$ is given by: $d(f,g)=\int^1_0 |f(x)-g(x)|\,\mathrm dx+\sup_{x\in[0,1]}|f'(x)-g'(x)|$ Now I aim to prove that: $C^1_0[0,1]$ with the matric $d$ is a complete metric space. My attempt: I need to pick a Cauchy sequence in $C^1_0[0,1]$ and […]

Can the product of two monotone functions have more than one turning point?

If we have 2 monotone functions $f$ and $g$ non zero, is it possible that $fg$ has more than one turning point. We can assume wlog that $f$ is increasing and $g$ is decreasing. $\frac{1}{x}e^x$ is an example of one turning point but I can’t think of any examples of more than one turning point. […]

Sum of uncountable many positive real numbers

Possible Duplicate: The sum of an uncountable number of positive numbers Suppose $f(x)>0$ for all real $x$, and $S$ is a set of uncountable many real numbers, how to prove that $\sum_{x\in S}f(x)=\infty$? Alternately suppose $\sum_{x\in S}f(x)=k$, how to prove $|S|=N_0$ ?

$f,g$ are both measurable on a set $ \Omega $, can $ \{ x\in \Omega: f(x)=g(x) \} $ be non-measurable?

Suppose $f,g$ are both measurable on a set $ \Omega $, can $ \{ x\in \Omega: f(x)=g(x) \} $ be non-measurable? My attempt: Let $\Omega$ be an open interval in $\Bbb{R}$, then it has a non-measurable subset, say $E$. Let $f=g$ on $E$, and $f>g$ one $\Omega \backslash E$. Is this ok? Thank you!