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By the choice of δ and because g e x y δ g x g y ²

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By the choice of δ and because g ∈ E , | x - y | < δ , | g ( x ) - g ( y ) | < ² 2 , thus g ( x ) < g ( y ) + ² 2 . By the definition of f ( y ) as a sup, g ( y ) f ( y ) . Putting it all together, we get f ( x ) < g ( x ) + ² 2 < g ( y ) + ² 2 + ² 2 f ( y ) + ² 2 + ² 2 = f ( y ) + ² ; that is f ( x ) - f ( y ) < ² . Changing the roles of x and y , we similarly get that f ( y ) - f ( x ) < ² , thus proving that | f ( x ) - f ( y ) | < ² if | x - y | < δ . Continuity follows. 4
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