# Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

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### Finishing the proof of the triangle inequality of Hausdorff metric

currently I am trying to show that a certain type of Hausdorff metric satisfies the following triangle equality and I am stuck. Setup: Take $(X,d)$ as metric space. Denote by $C(X)$ the set of closed ...
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### If $G\in C([0,1])$ and strictly increasing, can we find a sequence $G_n\n C^{\infty}$ with uniformly equicontinuous density?

Let $G:[0,1]\rightarrow [0,1]$ be a strictly increasing and continuous cdf with $G(1)=1$. I have proven some property for $G\in C^{\infty}([0,1])$ that relies on the continuity of $g(x)=G'(x)$. I hope ...
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### Can a function over a domain with holes have a derivative of zero everywhere but fail to be constant? [duplicate]

I've been reviewing some real analysis to prepare for topology and I got myself into a pickle while trying to understand disconnected sets and holes. Here is some reasoning that arrives at a weird ...
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### If $(X, A, m)$ is $\sigma$-finite and $B\subseteq A$ is a $\sigma$-sub-algebra, then is $(X, B, \nu)$ $\sigma$-finite?

Here $\nu$ is defined in the following way: $$\nu(E)= \int_E f dm$$ where $f$ is non-negative, $A$-measureable. I don't see why we can assert that $\nu$ is also $\sigma$-finite since the subsets of $B$...
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### Proof that the $\Bbb Q$ is dense in $\Bbb R$ (need check on intuition) [duplicate]

In this exercise, I am practising by trying to show that $\Bbb Q$ is dense in $\Bbb R$ I begin by stating I want to show that for any epsilon and for any $x$ in $\Bbb R$, there exists a rational ...
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### How to define double factorial for non positive integers?

I studied double factorial which known for natural number $$n!!=n(n-2)!! , 1!!=0!!=1$$ So we have for $n\in N$ $$(2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$ but I found on Math-World formula ...
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### Finding $\delta$ for convergence of $f(x)=x^3+2x^2-3x-1$ at its approximate root.

I've had some exposure to elementary analysis and I am currently going through a problem in numerical analysis involving finding roots using Newton's method. The algorithm has convergence issues which ...
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### Show that $\liminf(x_k \cdot y_k) = x \cdot \liminf(y_k)$ [closed]

We have a sequence $(x_k)$ in which $\lim(x_k)= x$ and $x > 0$. We also have that $\exists k_0$ such that $y_k > 0 \space \forall k \geq k_0$, then show that $\liminf(x_k y_k) = x \liminf(y_k)$. ...
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### Extension of a Hölder function on the product space

Let $k\geq0, \alpha>0$ and let $\Omega\subset \mathbb{R}^d$ be a $C^{k,\alpha}$ convex domain. Suppose $f:[0,1]\times \overline{\Omega} \to \mathbb{R}$ satisfies the following: For each $t\in[0,1]$...
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### Analytic way to find the sup of this sequence

I'm puzzled on how to find the sup of the sequence $a_n = n\cdot 2^n - n!$, without using tables of values and numerical calculations. Let me explain: by plotting the sequence, or by trying some ...
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### Why do we need the right-continuity of $F$ at $a$ to prove $V_F[a,c]=\lim_{b\to a^+}V_F[b,c]$ where $F$ is of finite variation?

This is a continuation of this post. The book claims the following without proof: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. Suppose also that $a<c$ and $F$...
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