No closest element to a closed subspace of $C([0,1])$ (Axler Example 8.27)

Question

There's just one claim in an example from Axler's Measure Theory text which I don't understand. See Example 8.27 below.

Analysis

So for any $g \in U$, it claims that $$ \tag{1} \int_0^1 (f-g) = \frac 1 2$$ and $$ \tag{2} (f-g)(1) = 0$$

implies that $||f - g || > \frac 1 2$.

I see that (note that I include absolute values) $$ \int_0^1 |f-g| = \frac 1 2 \le \sup_{[0,1]} |f-g| $$

which implies that $\text{distance}(f,U) \ge \frac 1 2$.

Question

But how does the condition in $(2)$ makes this inequality strict?