Articles of real analysis

Integration by parts in Riemann-Stieltjes integral from Rudin

Suppose $\alpha$ increases monotonically on $[a,b]$, $g$ is continuous, and $g(x)=G'(x)$ for $a\le x \le b$. Prove that $$\int \limits_{a}^{b}\alpha(x)g(x)dx=G(b)\alpha(b)-G(a)\alpha(a)-\int \limits_{a}^{b}Gd\alpha.$$ I thought on above problem more than two days and I think that I coped. I would be really greatful to anyone who check my solution! Proof: Let $\epsilon>0$ be given. Since $g\in C[a,b]$ […]

Inverse of Heine–Cantor theorem

We have by Heine–Cantor theorem that: If $M$ and $N$ are metric spaces and $M$ is compact then every continuous function $f : M \to N$, is uniformly continuous. Is the inverse of this theorem is satisfied?

Sum of sawtooth function not differentiable at dyadic rational points

This question is related to this question. Define $h(x)=|x|$ on $[-1,1]$ and extend it to $\mathbb R$ by defining $h(2+x) = h(x)$. This is a sawtooth function that is $0$ at even and $1$ at odd integers. Furthermore define $h_n(x) = (1/2)^n h(2^n x)$ and $$ g(x) = \sum_{n=0}^\infty {1\over 2^n }h(2^n x) = \sum_{n […]

Produce a sequence $(g_n):g_n(x)\ge 0$ and $\lim g_n(x)\neq 0$ but $\int_{0}^{1} g_n\to 0$

Produce a sequence $(g_n):g_n(x)\ge 0,\,\forall x\in [0,1],\,\forall n\in\Bbb N$ and $\lim g_n(x)\neq 0,\,\forall x\in [0,1]$ but $\int_{0}^{1} g_n\to 0$ Im in need to clarify that Im talking of the Riemann integral. I want some hint or example, Im unable to find a sequence like this. My work at this moment: If the integral converges to […]

Is the subset $ ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not compact as it can be expressed as union of the two disjoint open sets $[0,{\sqrt2}/{2}) $and$ ({\sqrt2}/{2}, \sqrt2)$ (though I’m not […]

limit of $\left( 1-\frac{1}{n}\right)^{n}$

limit of $$\left( 1-\frac{1}{n}\right)^{n}$$ is said to be $\frac{1}{e}$ but how do we actually prove it? I’m trying to use squeeze theorem $$\frac{1}{e}=\lim\limits_{n\to \infty}\left(1-\frac{1}{n+1}\right)^{n}>\lim\limits_{n\to \infty}\left( 1-\frac{1}{n} \right)^{n} > ??$$

Show $\sum\limits_{k=1}^{\infty} \frac {1}{(p+k)^2} = -\int_0^1 \frac{x^p \log x}{1-x}\,dx$ holds

Prove that $$\sum\limits_{k=1}^{\infty} \frac {1}{(p+k)^2} = -\int_0^1 \frac{x^p \log x}{1-x}\,dx$$ for $p>0$. I tried to transform LHS as Riemann sum form but failed. Can anyone give some idea? Many Thanks!

Computing fundamental forms of implicit surface

This question already has an answer here: About the second fundamental form 1 answer

Proving that the Union of Two Compact Sets is Compact

Prove if $S_1,S_2$ are compact, then their union $S_1\cup S_2$ is compact as well. The attempt at a proof: Since $S_1$ and $S_2$ are compact, every open cover contains a finite subcover. Let the open cover of $S_1$ and $S_2$ be $\mathscr{F}_1$ and $\mathscr{F}_2$, and let the finite subcover of $\mathscr{F}_1$ and $\mathscr{F}_2$ be $\mathscr{G}_1$ […]

Checking on some convergent series

I need some verification on the following 2 problems I attemped: I have to show that the following series is convergent: $$1-\frac{1}{3 \cdot 4}+\frac{1}{ 5 \cdot 4^2 }-\frac{1}{7 \cdot 4^3}+ \ldots$$ . My Attempt: I notice that the general term is given by $$\,\,a_n=(-1)^{n}{1 \over {(2n+1)4^n}} \,\,\text{by ignoring the first term of the given series.}$$ […]