When your baby is 5 weeks old, he is not able to stay awake for long periods. However, that does not mean he's going to sleep well for you, especially at night. At 5 weeks, he may begin understanding the difference between night and day, but he still may have trouble sleeping. If he's not getting enough sleep, it can become a problem for him and his parents, too.
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Monitor your baby's nap times during the day. You might not have your baby on a nap schedule yet, because babies are irregular sleepers at 5 weeks, but it is good to be aware of her tendencies. According to BabyCenter.com, 5-week-old babies generally can't stay awake for more than 2 hours at a time, and being awake for more than 2 hours can cause babies to get overtired and refuse to sleep. But do not let your baby nap for more than 3 hours because this can cut into her nighttime sleep and can also cause her to confuse nighttime and daytime.
Create a nighttime routine for your baby. It can help your baby understand that it is now nighttime and that he should go to sleep -- and stay asleep. Your routine can include a bath, a story, lullabies and time in the rocking chair. You do not need all of these things; your main concern is developing a routine you can stick with and do easily.
Look for signs that your baby no longer wants to be swaddled. Many babies outgrow swaddling by the time they are 2 months old, but at 5 weeks, your baby might want a little more freedom while he's sleeping. If you find your baby waking up with his swaddle considerably loosened or entirely broken, that could be a sign that he does not want to be swaddled anymore. Try giving him a lightweight blanket or a sleep sack to keep him warm. If he doesn't have a problem with being swaddled, continue to swaddle him, but keep an eye out in the future.
Let your baby cry it out within reason. Some babies get so attached to falling asleep while being held by their parents that they are scared to fall asleep on their own. Letting your baby cry for a few minutes can show her that it is okay to fall asleep without being rocked or held. By doing this, you're also helping her learn how to self-soothe, which can put her on the path toward sleeping through the night. If 10 minutes pass and your baby still has not put herself back to sleep, it is probably a sign that she needs something, such as a diaper or a feeding.