# Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

887 views

2k views

### Norm for pointwise convergence

Does there exist a norm on the space of all real-valued functions on the real line (or on an open set? a compact set?) such that convergence in this norm is equivalent to pointwise convergence?
508 views

436 views

9k views

### Density of irrationals

I came across the following problem: Show that if $x$ and $y$ are real numbers with $x <y$, then there exists an irrational number $t$ such that $x < t < y$. We know that $y-x>0$. ...
4k views

### Sine function dense in $[-1,1]$

We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by ...
1k views

### Maybe a rather famous integral

How to evaluate : $$\int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x$$ Thx guys! I was wondering how would use a series expansion?
1k views

### Finding the limit of a sequence with an undesirable $\ln k$

I am trying to compute the limit of this sequence: $$\lim\limits_{n \to \infty} \dfrac{(-1)^nn^2}{n!} \sum\limits_{k=2}^{n}\binom{n}{k}(-1)^kk^{n-1}\ln k$$ I can compute without the $\ln k$ in the ...
3k views

### If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
1k views

### Were “real numbers” used before things like Dedekind cuts, Cauchy sequences, etc. appeared?

Just the question in the title, I'm trying to understand how something like analysis could be developed without formal constructions of the real numbers. I'm also very interested, if the answer is "...
1k views

### A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$

What is the proof of the following: $$\int_{0}^{1} \left(\frac{\ln t}{1-t}\right)^2 \,\mathrm{d}t=\frac{\pi^2}{3} \>?$$
2k views

### Proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$

Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in this book. It has to do with using an inequality that relies on the binomial theorem. I tried to do an alternate proof ...
9k views

821 views

### Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$

How to compute the following integral $$\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$$ I have been given two integral questions by my teacher. I cannot answer this ...