 Missing-outcome bounds for a binary endpoint

Suppose an endpoint is binary, $N$ people are in the target cohort, $r$ observed outcomes are successes and $m$ outcomes are missing. With no assumptions about the missing values, the cohort success proportion $p$ satisfiesFor the proof, let $u$ be the number of successes among the missing outcomes. Since $0\le u\le m$ and $p=(r+u)/N$, both inequalities follow. Every integer $u$ in that range is a possible completion of the observed data, so the endpoints are sharp. These are missing data bounds on a cohort proportion, not confidence intervals or bounds on a causal treatment effect.