TITLE: The norm of the difference of two projections in a C$^\ast$-algebra
QUESTION [6 upvotes]: I'm trying to show that if $p$ and $q$ are projections in a
  $C^*$-algebra (i.e. $p=p^2=p^*$), then the norm of their difference is
  less than or equal to $1$.

Any help would be greatly appreciated!

REPLY [7 votes]: Let $p$ be a projection in $A$.  Recall that $\sigma(p)\subset\{0,1\}$.  Then in the unitization $A^+$ of $A$, by functional calculus we have 
$$\left\|p-\frac12\right\|\leq\max\left\{\left|0-\frac12\right|,\left|1-\frac12\right|\right\}=\frac12.$$
Thus, for any projections $p,q\in A$, we have
$$\|p-q\|\leq\left\|p-\frac12\right\|+\left\|\frac12-q\right\|\leq1.$$