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IMC 1995 Problem 12

Source: IMC 1995

February 19, 2021
functionreal analysis

Problem Statement

Suppose that (fn)n=1∞(f_{n})_{n=1}^{\infty} is a sequence of continuous functions on the interval [0,1][0,1] such that ∫01fm(x)fn(x)dx={1if  n=m0if  n≠m\int_{0}^{1}f_{m}(x)f_{n}(x) dx= \begin{cases} 1& \text{if}\;n=m\\ 0 & \text{if} \;n\ne m \end{cases} and sup⁡{∣fn(x)∣:x∈[0,1] and n=1,2,… }<∞\sup\{|f_{n}(x)|: x\in [0,1]\, \text{and}\, n=1,2,\dots\}< \infty. Show that there exists no subsequence (fnk)(f_{n_{k}}) of (fn)(f_{n}) such that lim⁡k→∞fnk(x)\lim_{k\to \infty}f_{n_{k}}(x) exist for all x∈[0,1]x\in [0,1].
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