Spectrum of difference of two projections Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ?
The exercise is to show that $\lVert p-q \rVert \leq 1$.
Any hints are appreciated.

You have, since $0\leq q\leq I$ and $0\leq p\leq I$, 
$$
-I\leq -q\leq p-q\leq I-q\leq I.
$$
So, as you mentioned, it follows that $\sigma(p-q)\subset[-1,1]$.
Note also that the argument does not use that $p,q$ are projections, only that they are positive elements of the unit ball.
