What procedure is offered at weekly visits after 37 weeks to help induce labor?

Sweeping membranes is the correct answer because this procedure is specifically designed to stimulate labor in pregnant women who are close to their due date, typically after 37 weeks of gestation. During a sweep, a healthcare provider inserts a gloved finger into the cervix and uses a circular motion to separate the membranes of the amniotic sac from the cervix. This action can release prostaglandins, which may help to soften the cervix and initiate contractions, potentially leading to labor.

The other options, while relevant in the context of pregnancy management, do not serve the specific purpose of inducing labor. A cervical check involves the examination of the cervix to assess its dilation and effacement but does not actively induce labor. Fetal monitoring is a procedure used to track the baby's heart rate and wellbeing, and it is crucial for assessing fetal health rather than inducing labor. Amniocentesis is a diagnostic procedure that involves taking amniotic fluid for testing potential genetic disorders or infections, but it is not performed for the purpose of inducing labor. Therefore, sweeping membranes stands out as the procedure aimed specifically at encouraging labor onset in the final weeks of pregnancy.