Solution: 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 \( n \to \infty \).

The best solution was submitted by 김범수. Congratulations!

Similar solutions are submitted by 권현우(+3), 박경호(+3), 오동우(+3), 이시우(+3), 이종원(+3), 이주호(+3), 장경석(+3), 장기정(+3), 정성진(+3), 정진야(+3), 조준영(+3), 채석주(+3), 한대진(+3), 황성호(+3). Thank you for your participation.

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3 thoughts on “Solution: 2014-01 Uniform convergence

  1. Fardad Pouran

    Please mention the deadline of each problem somewhere. Unfortunately I spent too much time for solving and TeXing the solution, but my work was done after the deadline.
    Best Regards.

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