Closure w.r.t $|| . ||_\infty$ Let $A=C[0,1]$ and $B=\{f\in A|f(0)=0\}$. What is the closure of B w.r.t the $|| . ||_\infty$ norm.
I really have a very poor intuitive grasp of this problem and I'm not sure where to begin?
I know $|| f ||_\infty=\sup_{x\in[0,1]} f(x)$ and that the closure is the limit points of B 
Thansk for any help

Alternatively:
Convergence in $\|\cdot\|_\infty$ is uniform convergence of functions, which implies pointwise convergence. Thus if there is a sequence in $B$ that converges to something outside $B$, then it would be a sequence of functions $f_n$ all with $f_n(0)=0$ which converged pointwise to an $f$ with $f(0)\ne 0$. This is clearly impossible, so $B$ is closed.
