# Tagged Questions

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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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### Proving a Cauchy Sequence without an explicit formula for $(a_n)$ [duplicate]

How would I prove that for all $n$ in the natural numbers, $|a_{n+1} - a_n| \le 1/2^n$ $(a_n)$ is a Cauchy Sequence? I have tried to use the standard definition and the triangle inequality, but ...
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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 ...