  An $[n,k]$ code $\mathcal{C}$ is said to be locally recoverable in the
presence of a single erasure, and with locality parameter $r$, if each of the
$n$ code symbols of $\mathcal{C}$ can be recovered by accessing at most $r$
other code symbols. An $[n,k]$ code is said to be a locally recoverable code
with sequential recovery from $t$ erasures, if for any set of $s \leq t$
erasures, there is an $s$-step sequential recovery process, in which at each
step, a single erased symbol is recovered by accessing at most $r$ other code
symbols. This is equivalent to the requirement that for any set of $s \leq t$
erasures, the dual code contain a codeword whose support contains the
coordinate of precisely one of the $s$ erased symbols. In this paper, a tight
upper bound on the rate of such a code, for any value of number of erasures $t$
and any value $r \geq 3$, of the locality parameter is derived. This bound
proves an earlier conjecture due to Song, Cai and Yuen. While the bound is
valid irrespective of the field over which the code is defined, a matching
construction of {\em binary} codes that are rate-optimal is also provided,
again for any value of $t$ and any value $r\geq3$.
