# Tagged Questions

For questions on infima.

0answers
25 views

0answers
21 views

### Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, ...
3answers
154 views

### Infimum and supremum of a set between 0 and 1

I am a little confused on what the infimum and supremum would be for the set S of all rational number between (0,1) not including 0 and 1. If 0 and 1 were included the answer is quite obvious.
1answer
41 views

### Infimum of absolute values versus absolute value of infimum

Let $A\subseteq\mathbb R$. Is there a nice proof of the inequality $\displaystyle\inf_{a\in A} |a|\le|\inf_{a\in A} a|$? The only proof I know is, though not very difficult, annoying because it ...
0answers
23 views

### Determine the Law of $F^{-1}(U)$

If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$) How can ...
3answers
80 views

### If $S=\text{{$-\infty,\ldots,0$}}$, does there exist infimum?

In this question What is the difference between minimum and infimum?, the answer of Thomas contains a line that It is a fact that every set of real numbers has ...
2answers
57 views

### Prove the infimum of monotone decreasing set equals its limit

Suppose that the sequence (sn) is monotone decreasing, in other words s1 ≥ s2 ≥ .... In addition suppose (sn) converges to s ∈ R. With these assumptions, prove that the set E = {s1, s2, ...} has an ...
1answer
64 views

### How to prove that the spectral radius of a linear operator is the infimum over all subordinate norms of the corresponding norm of the operator.

I am trying to understand a proof I have seen of the following theorem: $$\rho(A)=\inf_{\|\cdot\|}\|A\|.$$ I understand that to do this, the idea is to show that 1) $\rho(A)\leq\|A\|$ for any norm, ...
1answer
78 views

### $\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
3answers
162 views

### Prove that infimum (A)=0 and that supremum (A)=1 in the following set

$$A=\{\frac{n}{m}:m,n \in \mathbb{Z}^+, m>n\}$$ Now, I know that, as $n$ approaches $0$ from above and as $m$ approaches infinity, $\frac{n}{m}$ gets arbitrarily close to $0$, but my professor ...
2answers
66 views

### Determine whether the following subsets of $\mathbb{R}$ are bounded.

$A=\{x+\frac{1}{x}:x \in (0,\infty)\}$ $B=\{x^2+xy^2:-2 \leq x \leq 1, -1 \leq y \leq 1\}$ I understand the what it means for a set to be bounded above and below, but how would I go about proving ...
2answers
351 views

0answers
76 views

### How to obtain the infimum of this inequalities?

Let $A$ be the family of functions $f(z)=z+a_2z^2+\cdots$ that are analytic in unit disk $D:\{z:|z|<1\}$ and $S$ is the subfamily of functions that are univalent in $D$. $R(a)$ is the subfamily of ...
0answers
75 views

### closest point property of subset of Hilbert space - what are the conditions for existence of inf?

I'm proving the closest point property of a subset of a Hilbert space, ie: $$H$$ is a Hilbert space with a norm generated by the inner product and so on. $$h\in H$$ is a point in H $$M\subset H$$ M ...
1answer
2k views

### $\inf A = -\sup(-A)$

Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$ So far this is what i have ...
1answer
74 views

### Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
1answer
120 views

### Finding the supremum of the following set [duplicate]

I am stuck on the following problem: Let $P=\{x \in \Bbb R: x\ge 0,\sum_{n=1}^{\infty}x^{\sqrt n}< \infty\}$.Then what is the supremum of $P$? Can someone help me out by providing some ...
1answer
66 views

1answer
53 views

1answer
324 views

### Proof that the infimum of a given set $E$ is unique

Prove that a lower bound of a set might not be unique but the infimum of a given set is unique. Attempt: Consider some $E \subset \mathbb{R}$ such that $E \neq \emptyset$. $E$ is bounded below ...
3answers
512 views

### Supremum/Infimum proof

Assume that $\inf(A)>0$ and let $A'=\left\{\frac{1}{x} : x\in A\right\}$. I need to show that $\sup(A') = \dfrac{1}{\inf(A)}$. I think this is quite simple, $\sup(A')$ must be $\dfrac{1}{\inf(A)}$ ...