Let a, b be a nondegenerate, closed, bounded interval a) Prove that given any polynomial P on R and any 0, there is a polynomial Q on R, with rational coefficients, such that P(x) Q(x) for all x a, b b) Prove that the space C a, b (see Example 10 6) is separable

Question: Let [a, b] be a nondegenerate, closed, bounded interval. a) Prove that given any polynomial P on R and any > 0, there is

Let [a, b] be a nondegenerate, closed, bounded interval.
a) Prove that given any polynomial P on R and any ε > 0, there is a polynomial Q on R, with rational coefficients, such that P(x) - Q(x)\ < ε for all x ∈ [a, b].
b) Prove that the space C[a, b] (see Example 10.6) is separable.

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