2014-01 Uniform convergence

Let \( f \) be a real-valued continuous function on \( [ 0, 1] \). For a positive integer \( n \), define
\[
B_n(f; x) = \sum_{j=0}^n f( \frac{j}{n}) {n \choose j} x^j (1-x)^{n-j}.
\]
Prove that \( B_n (f; x) \) converges to \( f \) uniformly on \( [0, 1 ] \) as \( n \to \infty \).

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2 thoughts on “2014-01 Uniform convergence

  1. S. Oum

    No you don’t have to.
    You can take a picture.
    PDF file is preferred; if you use a smart phone, then there must be an app to make a PDF file by taking pictures.

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