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proving the theorems of limits of functions

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yes please. some hints would be good.

 

could u give me some cauchy sequence which does not converge.

 

[edit] oops x_n=a for all n is a cauchy sequence that doesnt converge rite[/edit] where d is the usual metric in R

 

[edit] dOH.' date=',,, that ovbisouly converges

 

it should be x_n = n[/quote']

 

No that is not a Caucy sequence. X_n = n does not stick together in the tail. R is a complete metric space so convergent sequence and Cauchy seqeunce are the same notion here. The notion of Cauchy sequence is a sequence where at the end of the sequence all elements are arbitrarely close on to the other.

 

Mandrake

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Some hints :

 

1) Uniqueness is quite trivial (Suppose there are two functions that satisfy the conditions, look at their difference and use the fact that for every point x in the closure of U there is a sequence entirely in U that converges to x, to show that both functions have to be equal. (Here you need the fact that the difference of two contniuous functions is again a continuous function)

 

2) => Take any point x in the closure of U and use the above fact about the sequence entirely in U that converges to x. Use uniformy continuity to conclude that f(x_n) is a Cauchy sequence in Y => hence convergent.

Define g(x) to be this limit.

 

3) Check that g is well defined and does not for instance depend on the choice of the sequence x_n above

4) Show that g is the same as f on U

5) Show that g is continuous.

 

Mandrake

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thanks a lot mandrake, i will have a stab at it sometime, i really need to brush on my l337 analysis skillz.

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