Q1. If 0<=a_1<=a_2<=a_3<=....,( 1,2,3 are the subscripts of a)
0<=b_1<=b_2<=b_3<=......(1,2,3 are the subscripts of b)
and a_n --> a and b_n -->b
Then prove that a_n*b_n -->a*b
Q2.Let f: R --> R be monotonically increasing, i.e.
f(x_1)<= f(x_2) for x_1< = x_2.
Show that f is measurable.
Hint: You may extend f to f':[-infinity,infinity]-->[-infinity,infinity]and
show that (x_alpha,infinity] is contained in f'^-1((alpha,infinity])
is contained in [x_alpha,infinity],where x_alpha=inf{x:f'(x)>alpha}
Please respond with Complete proof/solution.
I need this urgently.
NOTE: f' means a bar on the head of f |
Request for Question Clarification by
mathtalk-ga
on
13 Sep 2004 06:09 PDT
Hi, dicepaul-ga:
In Q1 it is unclear what kinds of things {a_n} and {b_n} are. From
the context I'd suspect these are sequences of functions, not
sequences of numbers, but you might want to clarify this. If
functions are intended, then we need to clarify what sort of
convergence is required here, e.g. pointwise convergence or some other
convergence property.
In Q2 the suggestion amount to looking at the inverse image of (alpha,
infinity] under f (or what is nearly the same thing here, f', since
the only arguments where f' is plus or minus infinity are resp. plus
or minus infinity). This is like the approach we discussed in the
earlier problem.
I'm afraid that the list price offered is not enough to allow me time
to do the quality of Answer that I would like.
[Google Answers: How to price your question]
http://answers.google.com/answers/pricing.html
Perhaps a better approach in the long run would be to find a tutor at
your University. Naturally I'd like to think Google Answers could
help, but really I suspect your Math. Dept. probably has a list of
people who are available.
regards, mathtalk-ga
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