# Tagged Questions

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### Distance between points in the plane [duplicate]

I have this problem and I honestly don't even have a clue of how to start, would someone help me please? Let $A$ = {$v_1$,$v_2$, . . . ,$v_n$} be a set of points in the plane such that the distance ...
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### Determine where do a function has limit.

I have to do the next exercise: Define $f:\mathbb{R} \to \mathbb{R}$ as follows: $$f(x)=x-[x]$$ if $[x]$ is even, and $$f(x)=x-[x+1]$$ if $[x]$ is odd. Determine those points where $f$ has a limit, ...
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### Prove that $f(x)= \frac{2x^{2}+3x-2}{x+2}$ has limit at (-2) and other exercises. [on hold]

I have to prove the next statements: 1)Define $f:(-2,0) \to \mathbb{R}$ by $f(x)= \frac{2x^{2}+3x-2}{x+2}$.Prove that $f$ has a limit at -2, and find it. 2)Suppose $f:D \to \mathbb{R}$ has limit at ...
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### Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, third edition, and am at Theorem 1.20(b), where he states and proves that between any to real numbers, there is a rational; that is, if ...
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### How to show that $\lim_{n \to \infty} \frac{\binom{n+k}{k}}{(n+k)^k} = \frac{1}{k!}$?

Given that $k\in\mathbb{N}$, my question is how to prove that this sequence converge to $\frac{1}{k!}$: $$\left\{ \frac{n+k \choose k}{(n+k)^{k}} \right\}_{n\in\mathbb{N}}.$$ I have this attempt:
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### Cauchy sequences.

Well my question is how to prove this: Let $a_{0}$, $a_{1}$ be distinct real numbers.Define: $$a_{n}=\frac{a_{n-1}+a_{n-2}}{2}$$ for each positive integer $n\ge2$.Show that $\{a_{n}\}$ is a Cauchy ...
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### An example of set with a countably infinite set of accumulation points

I have to give An example of set with a countably infinite set of accumulation points, and I say: We can consider the set or real numbers and we take an arbitrary real number $x$ then the interval ...
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### Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

For each real number $x$, let $[x]$ be the largest integer less than or equal to $x$. For example, $$[5] = 5$$ $$[7.9] = 7,$$ and $$[−2.4] = −3.$$ An arithmetic progression of length $k$ is a ...
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### Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
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### Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...
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### I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
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### Proving analytical statement, Intermediate Value Theorem

Let's define $f$ as a continuous function with $f:[0;2] \to \mathbb{R}$ and $f(0) = f(2)$. Now, I want to show that: $$\exists x_0 \in [0;1]:f(x_0) = f(x_0 + 1)$$ I tried to plot a few functions in ...
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### Proof of continuity - (ε-δ) definition - Can anyone check this?

I've been trying to get my head around this problem for quite some time by now. I want to prove that $$f(x) := \left|\frac{x-1}{x^2+1}\right|$$ is continuous for $$x_0 = -1$$ Now, in order to prove ...
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### Analysis: Prove the converse

It can be shown that if $\lim_{n\to\infty} a_n = L$, then $\lim_{n\to\infty} |a_n| = |L|$. Is the converse of this result true?
In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...