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Intro to Analysis in-class_Part_61

# Intro to Analysis in-class_Part_61 - 7.5 Approximation by...

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7.5 Approximation by polynomials 127 Step (3) If f C ( K ), x K , and ε > 0, then there is a function g x ∈ B such that g x ( x ) = f ( x ) and g x ( t ) > f ( t ) - ε , for t K . Proof. Note that x and ε are fixed. Since A ⊆ B and A satisfies the hypothesis of the previous theorem, so does B . Then for any fixed y K , we can find h y ∈ B that agrees with f at x and y : h y ( x ) = f ( x ) , h y ( y ) = f ( y ) . By the continuity of h y and the positivity theorem, there exists an open neigh- bourhood U y of y such that h y ( t ) > f ( t ) - ε, t U y . Since K is compact, there is a finite set of points y 1 , . . . , y n such that K U y 1 ∪ · · · ∪ U y n . Put g x := max { h y 1 , . . . , h y n } . By the last step, g x ∈ B , and g x has the required properties. Step (4) If f C ( K ) and ε > 0, there is h ∈ B such that | h ( t ) - f ( t ) | < ε , for t K . Proof. For each x K , we have the function g x ∈ B from the previous step. By the Positivity Thm, each x K has an open neighbourhood V x for which g x ( t ) < f ( t ) + ε, t V x .

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Intro to Analysis in-class_Part_61 - 7.5 Approximation by...

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