A question about projections in a C* algebra I have been struggling to prove the following result from a long time. The result is that for any two projections $p,q$ in a C* algebra $A$, we must have $||p-q||\leq 1$. This seems to me an astonishing result because if the strict inequality holds then there will alway exist a continuous path $\varphi:[0,1]\rightarrow \mathcal{P}(A)$ with the end points bein $p$ and $q$, which alone gives rise to many other interesting consequences. Any help will be highly appreciated.

The result actually holds for $p,q$ positive, they don't have to be projections. This was answered two days ago.