a problem between $C[0,1]$ subspaces. 
Given $X=\left\{x\in C[0,1], x(0)=0\right\}$ and $Y=\big\{y\in X, \int_{0}^{1}y(t)dt=0\big\}$, $\|x\|=\sup\{|x(t)|,t\in [0,1]\}$. Prove that, $\forall x\in X$ with $\|x\|=1 $  exists $y_0\in Y$ such that $\|x-y_0\|<1$.

I tried to prove by contradiction: $\exists x_0\in X, \|x_0\|=1, \|x_0-y\|\geq1, \forall y\in Y$. Then considering $d(x_0,Y)\geq 1$ and I tried to consider a sequence in $Y$... But I don't know how to finish it... I also tried to use Riesz's lemma (because $Y$ is closed) and try a sequence $x_n\in X$ that $\|x_n-y\|\geq 1-1/n$, but I think that doesn't help...
 A: Assume it's not true.Then there exists some $f$ in $X$ such that for every $g$ in $Y$ , $||f+g||\geq1$.Note,for now,that $Y$ is a closed subspace of co-dimension $1$.Let $S$ be the affine subspace $f+Y$.Our assumption would then imply that $S$ is a supporting hyperplane for the closed unit ball in $C[0,1]$.Consider the function $h$ in $C[0,1]$ given by $h(x)=1$ for all $x$ in $[0,1]$.Since,as I mentioned earlier,$Y$ is a closed subspace of co-dimension $1$,the line generated by $h$ intersects $S$,and since $S$ is a supporting hyperplane,the point of intersection lies on or outside the unit sphere.Let's take $k$ in $\mathbb{R}$ such that $kh$ belongs to $S$.Clearly $|k|\geq1$.Now there exists $g'$ in $Y$ such that $kh=f+g'$.Integrating both sides,and taking modulus,we have $|\int_{0}^{1}\hspace{1mm}f|\hspace{1mm}=\hspace{1mm}1$.Using triangle inequality,and the fact that $||f||=1$,we have $\int_{0}^{1}\hspace{1mm}|f|\hspace{1mm}=\hspace{1mm}1$.Now using the fact that $f$ is continuous and $|f(x)|\leq1$ for all $x$,we conclude that $|f(x)|=1$ for all $x$ in $[0,1]$,contradicting the fact that $f(0)=0$.
Edit - It might be helpful to visualize what happens in the finite dimensional case,that is,when you take all continuous maps on ${0,1}$ instead of $[0,1]$.This is nothing but $\mathbb{R}^{2}$ with the $l_{\infty}$ norm.The unit ball in this case is the square $[-1,1]^{2}$.Your subspace $Y$ corresponds to a line which is neither horizontal nor vertical.The subspace $X$ corresponds to the $y$ axis.Now imagine,if such a line,which happens to be oblique,is translated a unit distance along the $y$ axis,and the resulting affine subspace is named $S$,then can the entire square be contained in one of the closed half-planes on either side of $S$?You are encouraged to draw this.And yes,the diagonal point $(1,1)$ corresponds to the function $h$ in the solution above. 
