Comments on Problem 6.3.12
Homework
Math 555

1. We start with a continuous function f : [0,1] &rarr R.
   For each n &epsilon N, define &phi_n , g_n : [0,1] &rarr [0,1]
   by &phi_n (x) = xⁿ and g_n (x) = f( xⁿ) = [f &bull &phi_n ] (x) .
2. Next read about half of page 318 from our text: from the start of section 8.1 to the end of the page.
3. Does our sequence { g_n } above converge pointwise on [0,1] to a function g (x) = lim _n&rarr&infin g_n (x) ? Ok, to show me that you read my blurb-in', on your homework, answer this question yes (and explicitly say what g is) or no (and explain why not) or I have not idea.
4. Next read the whole of page 319. Part (c) says that it is not necessarily true that ∫₀¹   [ lim _n&rarr&infin g_n (t) ] dt    =    lim _n&rarr&infin [ ∫₀¹ g_n (t) dt ]    . So do not make this mistake when doing problem 12. It is kinda easy to see the difference between the right and left -hand sides above but when you get in the midst of a bunch of &epsilon's, be careful of the dependency as to not make this mistake.
5. Now do Problem 6.3.12. Another warning: we cannot assume, a prior, that the limit, as n&rarr&infin, of the sequence { ∫₀¹ f(xⁿ ) } from R exists.

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