## Question

###### Problem 1._ We say a function [0, 1] 3 [0. 1] is a line if there are m,b â‚¬ R such that f(x) =mc+b for all x â‚¬ [0,1]. Let L([0 , 1] 4 [0, 1]) denote the subspace of C([0,1] ~ [0,1]) with the metrie dx consisting of lines. Show L([O,1] ~ [0,1]) is compact: Hint: First, show that if In(x) mnx + bn, then there are subsequences (mnk JkeN and (bnre )keN that converge in R.

Problem 1._ We say a function [0, 1] 3 [0. 1] is a line if there are m,b â‚¬ R such that f(x) =mc+b for all x â‚¬ [0,1]. Let L([0 , 1] 4 [0, 1]) denote the subspace of C([0,1] ~ [0,1]) with the metrie dx consisting of lines. Show L([O,1] ~ [0,1]) is compact: Hint: First, show that if In(x) mnx + bn, then there are subsequences (mnk JkeN and (bnre )keN that converge in R.

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