7. Let [a, b] be a nondegenerate, closed, bounded interval.

(a) Prove that given any polynomial P on R and any f > 0, there is a polynomial Q on R, with rational coefficients, such that IP(x) - Q(x)1 < f for all x E

[a,b].

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(b) Prove that the space C[a, b] (see Example 10.6) is separable.