MandrakeRoot Posted November 25, 2004 Share Posted November 25, 2004 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 Link to comment Share on other sites More sharing options...
MandrakeRoot Posted November 25, 2004 Share Posted November 25, 2004 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 Link to comment Share on other sites More sharing options...
bloodhound Posted November 25, 2004 Share Posted November 25, 2004 thanks a lot mandrake, i will have a stab at it sometime, i really need to brush on my l337 analysis skillz. Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now