Real Analysis

[/0]

Then

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)

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]

Another question,

Am I right by saying that if a sequence is non-Cauchy, then it's divergent?

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)

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?

[/0]

The supremum of a set is the least possible upper bound of the set. The supremum need not be in the set.

The maximum of a set is a number which is an upper bound of the set, i.e. , . The maximum must be an element of the set.
As a result, some kinds of sets such as open sets don't have maximums. All finite partially ordered sets have maximums.

[QuantumJG]

What does the standard topology of a metric space mean?

[kamil9876]

The usual toplogy on a metric space is one where the open sets are those sets that are the unions of open balls, along with the empty set.

[dcc]

Approaches 2.25 as n is this right?

Not sure if its still relevant, but you should be able to recognise this sum as a relative of the Geometric series - Consider the derivative of .  (Have you learnt about uniform continuity and power series?)

edit: fixed sum index.