1
vote
0answers
55 views

Backward from infinity?

The following question has been raised and answered lately: Problem 6 - IMO 1985 Please take a look at the Reverse method part of the answer given by this author. What's happening there is that we ...
1
vote
3answers
32 views

Search for two Real Valued functions.

Can we have two real valued functions $f_1$ and $f_2$ defined on $[a,b]$ such that $f_1(x)=f_2(x)$ for infinitely many points and $f_1(x)\neq f_2(x)$ for infinitely many points. ?
1
vote
1answer
78 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
4
votes
2answers
158 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
0
votes
2answers
72 views

$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded? [duplicate]

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a differentiable function such that $\lim\limits_{x\to\infty}f'(x)=1$,then is it true necessarily true that $f(x)$ unbounded? I think that it will always ...
1
vote
1answer
58 views

Question about $f$ continuous function with these conditions?

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$ The question is: ...
2
votes
3answers
251 views

Why can't you count real numbers this way?

Sorry but this is probably a naive question. Why can't you generate real numbers by a*10^b, the same way as rational numbers by a/b? a and b could be integers so that you would start counting real ...
0
votes
2answers
59 views

Why does this limit work?

Let $h(x)= (1+1/x)^x$ and $g(x)$ be another function. Now suppose $\lim\limits_{x \to \infty} g(x)= \infty$. Then $\lim\limits_{x \to \infty} h(g(x))$ =$\lim\limits_{x \to \infty} h(x)=e$. I would ...
10
votes
7answers
325 views

A proof of a property of limits

Today during lecture my lecturer showed us this property, but provided no proof. If $$\lim_{n\to\infty} {d_{n+1}\over d_n} >1$$ then $$\lim_{n\to\infty}d_{n}=\infty $$ Is this property legit? ...
0
votes
3answers
154 views

What does the notion of different sizes of infinity really mean?

I have heard that there are infinities of various sizes. I was wondering what that actually means-how do we compare their cardinalities? I have just started real analysis and I am slowly coming to ...
1
vote
1answer
109 views

Why should the set have finite measure in the following proposition?

Here is a proposition in Royden: Assume $E$ has finite measure. Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise a.e. on $E$ to $f$ and $f$ is finite a.e. on $E$. ...
1
vote
1answer
238 views

Continuity and pushing a limit inside the function's domain

Consider some right-continuous function $f:\mathbb{R\cup\{-\infty,\infty\}}\to [0,1]$. I have to evaluate (i) $\lim_{b \to 0^+} f(\frac{a}{b})$, and (ii) $\lim_{b \to 0^-} f(\frac{a}{b})$ where $a \in ...
3
votes
2answers
1k views

There's a real between any two rationals, a rational between any two reals, but more reals than rationals?

The following statements are all true: Between any two rational numbers, there is a real number (for example, their average). Between any two real numbers, there is a rational number (see this proof ...
39
votes
3answers
4k views

Are there any series whose convergence is unknown?

Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will ...
2
votes
5answers
411 views

Does this expression represent the largest real number?

I'm not very good at this, so hopefully I'm not making a silly mistake here... Assuming that $\infty$ is larger than any real number, we can then assume that: $\dfrac{1}{\infty}$ is the smallest ...
4
votes
2answers
336 views

Is the number of circles in the Apollonian gasket countable?

Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar? What about ...
2
votes
2answers
484 views

Understanding of convergence of intersections of sets

If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of (-1/n, 1/n) as n->infinity)) However, at no time during the sequence is ...