Yes, a few species of bats eat bugs; but there are types of bats who only eat fruit.
However, bedbugs are not the only nocturnal feeder on humans. Quite a few other arthropods could be the cause of bite marks, including fleas, mites, spiders, or even bat bugs. Also, many medical conditions cause rashes that look similar to bug bites. If the marks persist but you don't find signs of an infestation, consider a trip to the doctor.
You have to figure out if you got them from your family or friends, church or from travelling. Eliminate the source of re infestation. You could be battling this for a very long time otherwise. Most likely if you pull up the mattress, you'll find bed bugs in the tufts of the seam of the mattress, as well as behind the tag of the mattress.
Bedbugs, about the size of an apple seed, lurk in cracks and crevices and feed on human blood. Though they don't transmit disease or pose any serious medical risk, the stubborn parasites leave
Harold Harlan, who was a career bug expert for the military, is a prominent authority on bed bugs. Below, he answered a few frequently asked questions: Are bed bug attacks a sign of dirty living
Yes its possible to only have a few bed bugs in a property. If you check on a regular basis and deal with them as you find them then its feasible to deal with them using the steps you have outlined already. In low levels infestations like this they can be easy to remove. I would suggest that you keep an eye on the situation and inspect once a week.