Q1. Let z_n, z be points in C and let d be the metric on C_infinity.
Show that |z_n - z| -->0 iff d(z_n,z)-->0.
Also show that if |z_n| -->infinity then {z_n} is cauchy in
C_infinity.(Must {z_n} converge in C_infinity).

Q2. Let {f_n} be a sequence of uniformly continuous functions
from(X,d)into (omega,gamma) and suppose that f= u-lim f_n exists.
Prove that f is uniformly continuous.If each f_n is a Lipschitz
function with constant M_n and supM_n<infinity, show that f is a
Lipschitz function.If
sup M_n=infinity,show  that f may fail to be Lipschitz(uniformly continuous).

Please provide the complete proof/solution for the above two problems.

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Request for Question Clarification by mathtalk-ga on 13 Sep 2004 06:27 PDT

Hi, dicepaul-ga:

In Q1 you mention C_infinity.  This is a notation that can have a
different meaning than the one I think you want here.  Please clarify
what C_infinity is and how the metric d on it is defined.  One might
think of C_infinity as a sphere that "completes" the complex numbers
by adding a point at infinity (often pictured as the "north pole"),
but the distance metric d can be defined in a number of (topologically
equivalent) ways.

Also you should clarify whether the final parenthetical phrase "Must
{z_n} converge in C_infinity" is meant as a question or as a simple
restatement of the previous sentence's conclusion.

In Q2's final part, think about how a uniformly continuous function
can fail to be Lipschitz.  Hint: Any continuous function on a closed
bounded interval is uniformly continuous, but if it become "very
steep" at a point, then it fails to be Lipschitz.  Avoiding this is
precisely the value of the assumption sup M_n is bounded in the
earlier part of this problem.

I will not be able to provide the detailed solutions you've requested
given the limited price offered.  However another Researcher may take
a special interest in these problems.  Normally a multiple-part
question would be listed at a substantially higher price, and as we
know you are likely to have a number of follow-up clarifications about
an answer to either part.

Please see my response to your other open question for some further suggestions.

regards, mathtalk-ga

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Clarification of Question by dicepaul-ga on 13 Sep 2004 06:36 PDT

Thanks for your suggestion about hiring tutor in my university.

Right now i just need the solutions/proofs, without explanation or clarification.

So even then if lot of time is required, i can compensate with the tip
part of the payment.