Just to clarify: With all due respect, Mary is not believed to have been "infallible" (unable to sin or make mistakes). She is, after all, human, and throughout her earthly life was surely tempted and quite able to sin.
Instead, we believe that Mary, "in the first instant of her conception, by a singular grace and privilege of Almighty God, and in view of the foreseen merits of Jesus Christ, the savior of the human race, was preserved free from all stain of original sin." We also believe that, with God's grace, Mary preserved this purity throughout her life and did not commit any actual sin. This doesn't mean that she could not sin (infalliblity), but that she did not sin.
Romans 3:23
For all have sinned, and come short of the glory of God;