All Questions
36
questions
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Partial proof explanation: Every bounded sequence in $R^k$ contains a convergent subsequence.
I am trying to understand the notation and implication of this part of a proof from Theorem 3.6 of Principles of Mathematical Analysis by W. Rudin:
Theorem:
Every bounded sequence in $R^k$ contains a ...
- 346
2
votes
1
answer
73
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Using the *compact* property of subsets in $R$ to prove Bolzano-Weierstrass Theorem
I have been asked to prove the Bolzano-Weierstrass Theorem with respect to a bounded sequence of real numbers by using the fact that closed and bounded subsets of $R$ are compact.
There is a hint ...
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1
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Are these statements true or false?
$\forall$ real $r \gt 0$, $\exists$ and natural number $M$ such that $\forall$ natural numbers $n>M$, $0 \lt \frac{1}{n} \lt r$
I think I understand this up until the last part $0 \lt \frac{1}{n} \...
- 329
2
votes
4
answers
743
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How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$?
Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$.
One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$...
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1
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2
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Real Analysis Proof: Corollary of Intermediate Value Thereom
I have recently picked up Patrick Fitzpatrick's Advanced Calculus : A Course In Mathematical Analysis and have come across a minor roadblock not too far into it.
Proposition 1.3: Let $c$ be a ...
- 720
1
vote
1
answer
559
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Issue with proof: Cauchy Completeness of Real Numbers
Having trouble understanding a cardinality-related argument when proving that all Cauchy sequences of reals numbers converge to a real limit. Came across it on CC Pugh's Real Mathematical Analysis, ...
- 10.1k