Consider $X=C[0,1]$ with its 'sup-norm' topology. Let $$S=\Bigg\{ f \in X: \int_0^1f(x)\;dx \neq 0\Bigg\}$$ How to prove $S$ is dense in $X$ ?
My try: I know this result " If a subset $A$ is nowhere dense in $X$, then $X \setminus A$ is dense in $X$"
The equivalent statement is " If $X \setminus A$ is NOT dense in $X$, then $A$ is NOT nowhere dense in $X$.
Here, $X \setminus S$ is a proper closed set, so it is not dense in $X$ and hence by above equivalent statement, $S$ is dense in $X$.
Is this right? Any help?