You asked

How can I tell if my child is getting all the nutrients that he needs?

The best that you can do is to try and ensure your child is benefitting from a varied diet and is getting foods from all the main food groups (grains, fruits and vegetables, dairy, meat and protein) at every meal.
It may not always be possible for you to get him to eat the amount that is recommended daily from each group every day of the week, but if you keep a healthy variety of foods available, he will get what he needs the vast majority of the time. 
One of the best ways to ensure a healthy balanced diet for your child is to get him to have a bowl of cereal with milk and a piece of fruit each morning for breakfast.

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