Consider the set of all continuous functions on [0,1], . Define . Prove that Proof so far. Let and As well as Then , there exists such that: So now, I have ... I'm stuck here... How should I go on from here? Thanks!
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Originally Posted by tttcomrader Consider the set of all continuous functions on [0,1], . Define . Prove that Proof so far. Let and As well as Then , there exists such that: So now, I have ... I'm stuck here... How should I go on from here? Thanks! if a+b = a -ε +b -ε +2ε then 0=0 For all xε[0,1] we have : sup{|f(x)-g(x)|:xε[0,1]} and sup{|g(x)-h(x)|: xε[0,1]} BUT: sup{|f(x)-g(x)|:xε[0,1]} + sup{|g(x)-h(x)|:xε[0,1]} = p(f,g) + p(g,h). Hence sup{|f(x)-h(x)|:xε[0,1]} p(f,g) + p(g,h) , thus p(f,h) p(f,g) + p(g,h)
But does necessarily takes on one of the value of for some ? If the sup is outside of this set, then the inequality doesn't hold. Thanks!
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