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**aliceinwonderland** Let $\displaystyle B[a,b]$ denotes the collection of all bounded functions from a closed interval [a,b] into R. Let $\displaystyle \rho$ be a metric for $\displaystyle B[a,b]$ such that $\displaystyle \rho(f,g) = lub \{ |f(x) - g(x) | : x \in [a,b]\}$.

Let $\displaystyle C[a,b]$ denotes the collection of all continuous real-valued functions on [a,b] such that $\displaystyle C[a,b]$ is a closed subspace of $\displaystyle B[a,b]$.

Let $\displaystyle \rho'$ be a metric for $\displaystyle C[a,b]$ such that $\displaystyle \rho'(f,g) = lub \{|f(x) -g(x) |: x \in [a,b] \}$.

(a) Show that $\displaystyle C[a,b]$ is not a compact in $\displaystyle B[a,b]$.

(b) Show that $\displaystyle C[a,b]$ is no where dense in $\displaystyle B[a,b]$.

(c) A sequence $\displaystyle \{f_{n}\}_{n=1}^{\infty}$ in $\displaystyle B[a,b]$ converges to a member $\displaystyle f \in B[a,b]$ with respect to the metric $\displaystyle \rho$ if and only if $\displaystyle \{f_{n}\}_{n=1}^{\infty}$ converges to f uniformly.