Posted by J. Holmer on October 28, 2007, 9:23 pm, in reply to "proof of Stone-Weierstrass"
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We used the hypothesis that A includes the constant functions when we showed that for any x_1\neq x_2 in X and a_1\neq a_2 constants, there exists f\in A such that f(x_1)=a_1 and f(x_2)=a_2. This was proved by using the fact that A separates points to get some h\in A such that h(x_1)\neq h(x_2), and constructing f=c_1h+c_2 by solving a 2x2 linear system of equations for c_1 and c_2. But note that we know f\in A since 1\in A and f is a linear combination of h and 1. 148
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