Example circumference metric space non complete

Question

I am asked to show using an example that space

where

is not a complete space.

I know the circumference is complete, but I don't understand why the other part makes it non complete.

Answer 1

Hint: You can find a sequnce which approximates a discontinuous function.

You can define a sequence of continuous functions which approximate $1_{[\frac{1}{2},2]}(x)$.

For example, $f_n(x)=(x+\frac{1}{2})^n1_{[0,\frac{1}{2}]}(x)+1_{(\frac{1}{2},1]}$.