Let $C[0,1]$ represent the set of all continuous functions on $[0,1]$. Is the following subset of $C[0,1]$ complete with respect to the metric $\rho(f,g) = \sup_{x \in [0,1]}|f(x)-g(x)|$?
The subset in question is $$X = \{f \in C[0,1] | f(x) > 1 \forall x \in [0,1]\}$$
I understand that to prove this we need to show that every Cauchy sequence converges to a limit in the metric space, but I am unsure how to do this.
It is not complete, since it is not closed. And it is not clolsed because, if you define $f_n\in C\bigl([0,1]\bigr)$ by $f_n(x)=1+\frac1n$, then $(f_n)_{n\in\mathbb N}$ converges in $C\bigl([0,1]\bigr)$ to the constant function $1$, which doesn't belong to $X$. However, each $f_n$ belongs to $X$.