Muddiest Point 11/14

For me, the muddiest point of this lecture was understanding the order of choosing N and \delta. I will illustrate this in the proof of Theorem 6.2.6.

Statement of Proof: If f_n \to f uniformly on A and f_n continuous on A for all n, then f is continuous on A as well.

Proof: Let \epsilon >0. Choose N such that ||f_N-f||_{\infty,A} < \frac{\epsilon}{3}. Choose \delta such that |x-c| < \delta implies |f_N(x)-f_N(c)| < \frac{\epsilon}{3}.. By the triangle inequality, |f(x)-f(c)| \leq |f(x)-f_N(x)|+|f_n(x)-f_n(c)|+|f_n(c)-f(c)| < 3 (\frac{\epsilon}{3})= \epsilon.

It is necessary to choose N before \delta because \delta relies on N. Imagine trying to choose \delta first. You wouldn’t be able to choose a \delta that would help solve the problem because the correct \delta relies on N.

Posted on by Grace Conway
Categories: Daily Blogs, Muddiest Point

One Response

  1. Jeremy LeCrone says:
    November 26, 2017 at 12:54 pm

    Good post. Thank you

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