Uni Stuff > Mathematics
Real Analysis
/0:
--- Quote from: Pappa-Bohr on March 28, 2010, 08:21:57 pm ---
--- Quote from: /0 on March 28, 2010, 06:51:12 pm ---Then
--- End quote ---
how do you get this? shouldn't there be a 'less than or equal to' sign instead of the equality? (i.e. the triangle inequality)
--- End quote ---
I don't know if you're learning about metric spaces, but in general, a metric is similar to the notion of 'distance' between x and y.
If we use the standard euclidean metric . This essentially is the 'distance' between two numbers, and it is what is normally used in sequences.
Thus we wish for a cauchy that for , :
For
Then (assuming of course ).
And since we can set , etc etc we see that it is not cauchy
QuantumJG:
Thanks /0. In a tute yesterday I was told what those things meant.
kamil9876:
--- Quote from: /0 on March 28, 2010, 07:57:06 pm ---
--- Quote from: QuantumJG on March 28, 2010, 07:27:44 pm ---Another question,
Am I right by saying that if a sequence is non-Cauchy, then it's divergent?
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Hmm, I think so:
Theorem 8.1.3: A sequence in converges (to a limit in ) iff it is Cauchy.
(the proof is nearly 2 pages)
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The proof of that can be made simple to the point that you can recreate it yourself, if you just split it into little pieces and treat it as a sequence of little lemmas, e.g: prove for first and have as just a corollary.
I'm gonna make this conjecture now that this thread got me thinking(I've been in the mood of making conjectures lately, so far half are true): Suppose are complete metric spaces, then the metric space with metric is complete.
Pappa-Bohr:
wat did u get for sheet 1 section 7 question 15.
i.e ''limit as n goes to infinity'' for 1+(-1)^(n+1)
(also is n assumed to be a positive integer or a real or what?)
QuantumJG:
What is the difference between the maxima of a set and the supremum of a set?
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