Metric spaces - continuity / compact

Hi, I'm doing the following assignment:

We consider the metric space M = C([0,1], R) of continuous, real functions on [0,1]. M has the usual uniform metric:

d_M(f,g)=sup{lf(x)-g(x)l l x E [0,1]}

(a) Show that I:M-->R given by:

I(f) = integral (from 0 to 1) f(x)*dx

is continuous.

(b) Show that I from (a) attains a max and a min on the closure of the unit ball:

Closure of K(0,1) = {f E M l d_M(0,f) =< 1}

Here 0EM is the nil-function and I don't have to show that the closure of the unit ball is as suggested.

(c) Show that f_n E M given by

f_n(x) = x^n, x E [0,1]

for n=1,2, ... defines a series (f_n) in the closure of K(0,1) that does not have a convergent sub-series in M and conclude that the closure of K(0,1) is not compact.

In (a) I think that an argument regarding I^(-1)(f) is sufficient. But I guess just saying that I^(-1)(f) = M, which is a closed set is too superficial?

In (b) the problem is that the closure of K(0,1) is not compact and hence I cannot use the usual theorems...

Any suggestions will be deeply appreciated... :-)