Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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10
votes
1answer
424 views

Infinity-to-one function

Are there continuous functions $f:I\to S^2$ such that $f^{-1}(\{x\})$ is infinite for every $x\in S^2$? Here, $I=[0,1]$ and $S^2$ is the unit sphere. I have no idea how to do this. Note: This is ...
2
votes
2answers
288 views

Where is the flaw in my Continuum Hypothesis Proof?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis (CH) has gripped my attention today, and I even asked a question about it earlier today. ...
2
votes
1answer
87 views

Is this interpretation of the continuum hypothesis correct?

I am not a mathematician, but rather a computer engineer with a curious mind. The continuum hypothesis states (I believe) that there does not exist a set $S$ such that $\aleph_0 < |S| < ...
0
votes
2answers
56 views

Unique infinite subsets of the integers

Edit: Great points on the comments. There is no unique set of unique infinite subsets of the integers. Is this a better question? What is the largest possible cardinality of a set which is a set of ...
27
votes
3answers
943 views

What is the largest set for which its set of self bijections is countable?

I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that ...
1
vote
1answer
90 views

Limit to infinity and infinite logarithms?

When trying to evaluate$$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$I noticed that the answer was bound to be complex for any $x$. Plugging in a very, very large real number in for $x$ will eventually ...
0
votes
1answer
208 views

Can I have something larger than infinite? [duplicate]

My question is "Can I have something larger than infinite?" Sometimes, we add infinite numbers into our set of numbers by simply extending our set and adding infinite numbers to it. But can't you ...
-2
votes
2answers
100 views

How small is infinite? [closed]

There are a lot of posts concerning how big infinite is, but I wonder how small infinite is. One can clearly see (ignoring a few things) that$$\frac{\infty}2=\infty$$Which means that no matter how ...
0
votes
1answer
41 views

A box comprised of infinite number of small similar boxes.

On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions" I don't understand what does Wikipedia wants to say with a box containing ...
4
votes
3answers
98 views

The cardinality of Indra's net?

This question has been asked before, but the title of the post was so general that it received no sufficient answer. What is the cardinality of the set of jewels and reflected jewels in Indra's Net? ...
1
vote
3answers
39 views

Classify the type of discontinuity at $x_0 = 0$

for (a) I think it is essential because the right side goes to infinity. for (b) I think it is removable because the function is not defined in $0,$ same goes for (c) I am really not sure about ...
0
votes
0answers
24 views

What is the origin of the distinction between assignable and inassignable number?

Leibniz described his infinitesimals as being inassignable numbers in a number of texts, e.g., in his Cum Produisset that was analyzed in detail by H. Bos in a seminal text dating from the 1970s. The ...
0
votes
3answers
290 views

Concept behind the limit to infinity?

I can across transfinite numbers and came up with a thought. What if$$\lim_{x\to\infty}f(x)=f(T)$$where $T$ was a transfinite number? Generally, in calculus, I have noted that it is two different ...
2
votes
1answer
56 views

Prove the squared vector 2-norm is $\leq$ sum of 1-norm and infinity-norm

How do I prove that $$\|x\|_2^2 \leq \|x\|_1 \|x\|_\infty?$$
2
votes
2answers
35 views

values that can be attained by random variables

Can a discrete random variables takes the values $+ \infty$ and $- \infty$ ? Can someone explain to me this with an example?
2
votes
2answers
78 views

Can I assume the continum hypothesis in a proof

I am showing that the cantor ternary set has the same cardinality as $\mathbb{R}$ I want to use the fact that it is uncountably infinite and a subset of $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...
-1
votes
2answers
79 views

Limit laws when not both limits exist

In the calculus textbooks I've come across, the limit laws are given on the condition that both individual limits exist. Is it safe to weaken that condition by saying that they are valid as long as ...
2
votes
1answer
77 views

There are more languages than programs?

I am reviewing some Turing machine material...and I come across this the set of all programs are countable (convert them into binary string, each of which represent an integer) whereas the set of ...
3
votes
1answer
44 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, ...
1
vote
3answers
74 views

Proof using formal definition: Infinite limit

I was wondering how get the proof of this limit: $$\lim\limits_{x\to -\infty}\dfrac{{x^2} - x + 1}{x + 4} = -\infty$$ The problem is that I don't know what to do for find the appropriated values to ...
0
votes
3answers
79 views

probability on countable infinite sets

My question relates to probabilities on countable infinite sets. For example, what is the probability of choosing an even number from the positive integers. Believe it or not I am interested in this ...
0
votes
1answer
57 views

Thomson's Lamp Question

The Thomson's Lamp paradox: A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a ...
0
votes
3answers
63 views

calculate the limit of the following function.

$$\lim_{x \to 0} \frac{5x - e^{2x} + 1}{3x + 3e^{4x} - 3}$$ For some reason I can't find the trick to solve this. Tried a lot but it always come to a place where its $\frac{0}{0}$ or ...
0
votes
0answers
49 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + ...
-2
votes
1answer
35 views

Zero-infinity hypothesis [duplicate]

math.stackexchange community. I have joined to inquire on a hypothesis a friend of mine has recently proposed. Please note: before posting this, I have repetitively told him that his logic is flawed ...
0
votes
2answers
34 views

size of infinite strings and infinite alphabets

Please forgive the lack of formal vocabulary. Which set has a larger cardinality? A) a set of all possible countably infinite strings with a finite alphabet of symbols. B) a set of all possible ...
0
votes
2answers
52 views

The intersection of an infinite number of subspaces is a subspace

Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite ...
0
votes
0answers
49 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
5
votes
4answers
125 views

Is $\aleph_1\cdot\aleph_1=\aleph_1$?

I'm currently trying to understand the basic notions concerning infinity. I think I understand that $\aleph_0\cdot\aleph_0=\aleph_0$ but how about $\aleph_1$? Is $\aleph_1\cdot\aleph_1=\aleph_1$ i.e. ...
4
votes
2answers
442 views

Compute a limit or prove that it does not exist

Do the following limits exist? Compute them or prove that they do not exist. (a) $\lim_{x\to 1}\frac{x^2-x}{2x^2-x-1}$ (b) $\lim_{x\to 1}\frac{|x-1|}{2x^2-x-1}$ For (a) it's pretty easy to see that ...
2
votes
2answers
38 views

Proving the following limit statements

I need to prove those If $f(x)\ge 0$ and $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}\sqrt{f(x)}=\sqrt{L}$. If $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}|f(x)|=|L|$. If $f(x)\ge g(x)$ for ...
1
vote
0answers
37 views

Friend B and C have eaten zero apples. How many more apples has C eaten?

Friend $A$ claims that he has eaten $1$ apple today. Friend $B$ responds. Congrats, I have eaten $0$ apples, so that is $\infty$ more apples than me. Friend $C$ says, but I have also eaten $0$ ...
1
vote
1answer
26 views

Probability of selecting a number in a repeating decimal series

For example in a infinitely repeating series such as $\frac{110}{111}=0.\overline{990}$, what would be the probability of selecting a 0 in the series generated by the infinitely repeating decimals? I ...
-2
votes
2answers
50 views

Limit of infinity times 0

I have a question regarding a specific step in the proof of the theorem that 'differentiability implies continuity'. The proof in my calculus book asserts that if $h\to0$ then: ...
1
vote
2answers
153 views

Is it true that the sum of all numbers equal 0? [closed]

I'm not a mathematician but I'm studying Nothing, so 0 is relevant, and I'm wondering about the fact that numbers seem to be mutually canceling polarities extending from 0, that is ...
0
votes
2answers
38 views

Determine limit of indeterminate form.

If the question is $$\lim_{x\to\infty}(e^x+1)^{\frac1x}$$ Do you just say that because $\lim_{x\to\infty}\frac1x$ is $0$, the original function has limit approaching 1, without caring the $e^x$?
2
votes
2answers
194 views

Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional.

Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional. It seems intuitively obvious that the dimension of ...
0
votes
0answers
32 views

How to solve a particular indeterminate form

So the answer says $$\lim_{x\to \infty}x^2\sin\left(\frac1x\right)=\lim_{h\to 0^+}\frac1h\frac{\sin h}h$$ How does the transformation work?
3
votes
2answers
43 views

Limits and infinity in a succession

Apologies for this rather basic question. I am preparing the entry exam for university without the help of a teacher and occasionally get stuck on seemingly simple things. I have been all over the ...
1
vote
1answer
41 views

Proving using the squeezing theorem

The question is : I am not sure how i should make the function bigger and smaller in order to find the right limit for it. Thanks in advance !
1
vote
0answers
37 views

Complex residue at infinity of $f(z)=\frac{z^5}{\sin\left(\frac{1}{z^2}\right)}$

I'm having trouble finding residue of the function $$f(z)=\frac{z^5}{\sin\left(\frac{1}{\large{z^2}}\right)}$$ at infinity. Wolfram kindly informs that it is equal to $-\frac{7}{360}$ (and gives ...
1
vote
1answer
19 views

mass concentration inequality for polynomials

I am trying to prove the following: Let $p$ be a polynomial of degree n and let $I=[0,1]$ and $E\subset I$ a measurable set of non-zero measure, i.e., $\mu(E)\neq 0$. Then, $$\sup_{x\in I}|p(x)|\leq ...
0
votes
2answers
82 views

Number of elements is odd infinity [closed]

If we have one continuous function F(x), and if we define f(x)=F(x) on domain from open interval (a, b), and if F(a)=F(b) If function f(x) is monotonically increasing from point a to point M, and ...
-4
votes
1answer
49 views

Function approaches zero but derivative doesn't [duplicate]

If: $y=f(x)$ and $y=0$ when $x\rightarrow\infty$ Is it possible that: $\frac{d}{dx}(y)$ is not equal to zero when $x\rightarrow\infty$ And prove it!
-1
votes
2answers
87 views

Showing that a set is countably infinite by defining a bijection between $\Bbb N$ and that set.

I'm a little confused on what is being asked here: Show that the following sets are countably infinite, by defining a bijection between $\Bbb N$ (or $\Bbb Z^+$) and that set. The set of positive ...
0
votes
3answers
33 views

proving that delta exists to a limit

The question is : Assume $\lim_{x\rightarrow1}f(x)=5$. Prove that there exists $\delta>0$ s.t. for every $x$ that sustains the condition $|x-1|<\delta$, we know that $f(x)>-1$. I know ...
1
vote
2answers
71 views

Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
2
votes
1answer
29 views

Linear independence and ∞-dimensioned vector spaces

I have a question regarding this problem: Let ℝℕ be the vector space of all infinite real sequences. Show that even though its infinite subset X≔{(1,0,0,…), (0,1,0,0,…), (0,0,1,0,…)} is linearly ...
1
vote
1answer
38 views

Once we've constructed N, do we need again the axiom of infinity to conclude the set of the primes is infinite?

When completing the proof (unfortunately, at the same time presenting it as Euclid's and performing it ad absurdum) of the infinitude of prime numbers, my algebra professor stated "...and thus the set ...
2
votes
3answers
78 views

How do I compute $\sum_{k=1}^{\infty} k \cdot p^k$ [duplicate]

I have no idea how to compute this infinite sum. It seems to pass the convergence test. It even seems to be equal to $\frac{p}{(1-p)^2}$, but I cannot prove it. Any insightful piece of advice will be ...