  Using proof-theoretical techniques, we analyze a proof by H.-K. Xu regarding
a result of strong convergence for the Halpern type proximal point algorithm.
We obtain a rate of metastability (in the sense of T. Tao) and also a rate of
asymptotic regularity for the iteration. Furthermore, our final quantitative
result bypasses the need of the sequential weak compactness argument present in
the original proof. This elimination is reflected in the extraction of
primitive recursive quantitative information. This work follows from recent
results in Proof Mining regarding the removal of sequential weak compactness
arguments.
