Real Analysis - compact and complete

If X is compact prove that C(X,R) is a complete metric space.

Real Analysis - In this space is every closed and bounded set compact?

Let (X,d) be the metric space consisting of m-tuples of real numbers with metric d(x,y)=max{|a_k-b_k|:k=1...m} where x={a_1, a_2,...,a_m} and y={b_1, b_2,...,b_m}. In this space is every closed and bounded set compact?

Real Analysis - Show E is equicontinuous.

Let E be a set of differentiable functions in C[a,b] with uniformly bounded derivatives; i.e., there exists a number M, independent of f in E, such that |f'(x)|<=M for all x in [a,b] and all f in E. Show that E is equicontinuous.

Real Analysis - contraction is continuous

Show that a contraction is continuous.

Real Analysis - Newton's Method show convergence

Newton's Method: Consider the equation f(x)=0 where f is a real-valued function of a real variable. Let x_0 be any initial approximation of the solution and let x_(n+1)=x_n - (f(x_n)/f'(x_n)). Show that if there is a positive number "a" such that for all x in [x_0-a, x_0+a] |(f(x)f''(x))/((f'(x))^2)|<=lambda<1 and |(f(x_0))/(f'(x_0))|<=(1-lamda)a then the sequence (x_n) converges to a solution of f(x)=0.