Let Y⊆ X be a closed subspace. By a simple argument, we show that Y⊥ ⊥⊆ X∗ ∗ is strongly proximinal at points of X if and only if Y is a strongly proximinal subspace of X. This substantially improves the main result of Jayanarayanan and Paul (J. Math. Anal. Appl.426 (2015) 1217–1231). As a consequence we get an easy proof of a classical result of Alfsen and Effros (Ann. Math.98 (1972) 98–173), that M-ideals are proximinal subspaces and a result of Dutta and Narayana (Function Spaces, Contemporary Mathematics, vol. 435, American Mathematical Society, Providence (2007) pp. 143–152), that M-ideals are strongly proximinal subspaces. © 2021, Indian Academy of Sciences.