Let $X$ be a subspace of the real space $C[0,1]$ consisting of all continuous functions $x$ on $[0,1]$ such that $x(0)=0$. For $X_0$ we take the subspace of all $x\in X$ such that $\int\limits_0^1 x(t) dt=0$. Now suppose that $x_1\in X$, $||x_1||=1$, and $||x_1-x||\geq 1$ if $x\in X_0$. Corresponding to each $y\in X\backslash X_0$, we let
$$c=\frac{\int\limits_0^1 x_1(t)dt}{\int\limits_0^1 y(t)dt}$$
Then
$$x_1-cy\in X_0$$.
How does this last line come about?
It's part of an example from the Functional Analysis text by Taylor.