All Questions
6
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Proof: For any subsequence $a_{n_k}$ Prove $\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n$
For any convergent subsequence $a_{n_k}$ of $a_n$, Prove: $$\liminf_{n\to\infty} a_n \le \lim_{k\to\infty} a_{n_k} \le \limsup_{n\to\infty} a_n.$$
My attempt
For this proof it should be noted that $...
- 70
-1
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1
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142
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How to prove given sequence converges to 0? [duplicate]
I am given a sequence {$a_{n}$} of nonzero numbers converges to infinity. How can I use this to prove that the sequence {$\frac{1}{a_{n}}$} converges to 0?
I can intuitively see why $\frac{1}{\infty}$...
- 79
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1
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Problem in proof of -"A net has $y$ as a cluster point iff it has a subnet which converges to $y$"
Directed Set:
We say that $(\omega, ≤)$ is a directed set, if ≤ is a relation on $\omega$
such that
(i) x ≤ y ∧ y ≤ z ⇒ x ≤ z for each x, y, z ∈ $\omega$;
(ii) x ≤ x for each x ∈ $\omega$;
(iii) for ...
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1
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0
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How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
How to prove that, for any sequence $(s_n)$ of real number and any real number $z$, the following $2$ statements are equivalent?
$1.$ Every subsequence of $(s_n)$ has a further subsequence that ...
- 2,758
0
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0
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20
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Help proving $ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
I am trying to formally prove:
$ n > \frac12 \frac xy | n \le \frac xy \lt n + 1, \forall n $
where n is an integer, and x and y are natural numbers.
It is obvious that, when $\frac xy$ is ...
- 101
2
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1
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How to construct binary sequences associated to points of the Cantor set?
Let $C\subset \mathbb{R}$ be the Cantor set, obtained from the interval $[0,1]\subset \mathbb{R}$ by removing the middle thirds of successive subintervals. That is, assuming $C_n$ constructed we let $...
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