Homework due Jan. 23 — MTH 429H

Homework due Jan. 23:

From Rudin Ch. 7: Exercises 3, 6, 11, 12. Additional exercises:

  1. (Exercise 9 from Rudin rephrased) Let (f_n)_{n=1}^\infty be a sequence of continuous functions on a set f.
    1. Prove that if f_n converge uniformly to f then

      \displaystyle{\lim_{n\rightarrow\infty} f_n(x_n)=f(x)}\ \ \ \ \ (\star)

      for every sequence of points x_n\in E such that x_n\rightarrow x and x\in E.

    2. Is the converse true?  That is, if (\star ) holds whenever x_n\rightarrow x does it follow that f_n converge to f uniformly?
  2. Use Theorem 7.11 from Rudin to prove the following result stated in class on Jan. 11:

    Let (a_{m,n})_{m,n=1}^\infty be a doubly indexed sequence such that

    \displaystyle{\lim_{n\rightarrow \infty}} a_{m,n}=b_m

    uniformly in m , i.e., for every \epsilon >0 there is an N   so that for n\ge N we have |a_{m,n}-b_m|<\epsilon for all m . If \displaystyle{\lim_{m\rightarrow \infty}} a_{m,n}=A_n for each n then (A_n)_{n=1}^\infty converges and

    \displaystyle{\lim_{m\rightarrow \infty}} b_m = \displaystyle{\lim_{n\rightarrow \infty}} A_n.

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