Is the space $(X,d)$ complete? If not what is its completion?
a) X is the set of all continuous on [0,1] functions, $$d(x, y) = \sup_{0 \le t\le1}\ t^2 |x(t) - y(t)|$$
b) $X = \{x\in C[0,1]: \sup_{0 \le t\le1}\ t^{-2} |x(t)| < \infty\}$, $d(x,y) = \sup_{0 \le t\le1}|x(t) - y(t)|$
For the solution of part a I came up with: Let $(x_n)$ be a Cauchy sequence and let $\epsilon>0$ be arbitrary. Then $\exists P \in \mathbb{N}$ such that $$d(x_n,x_m) < \epsilon\ \forall n,m\ge P$$ $$\sup_{0 \le t\le1}\ t^2 |x_n(t) - x_m(t)|< \epsilon\ \forall n,m\ge P$$ Then, $\sup_{0 \le t\le1}\ t^2 |x_n(t) - x_m(t)| \le \sup_{0 \le t\le1}\ |x_n(t) - x_m(t)|$
But then I do not know what to do next I do not know if such an approach is true: r > 0 be arbitrary, $ \sup_{0 \le t\le1}\ |x_n(t) - x_m(t)| < r $
thus metric space is complete.
Also for the solution of part b I thought I would follow almost same steps but cannot find a Cauchy sequence that shows it is not complete.