Prince Harry wanted Meghan to meet the Royal children so that he could show her what their life would look like once they have kids. Harry especially wanted Meghan to see how normal the children are and how regularly they are being brought up, to prove to her that Royals too can have regular childhoods.
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Harry is great with kids and according to their biographer, one of the reasons why Meghan fell for Harry was the fact that he was so easy and fun with children around, and that gave her a lot of hope.
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Coming back to the Queen. The Queen actually opened up about how Royal children can be more careful. She pointed out that the trick to keeping children in line is to let them put up the decorations so they are careful about knocking them down.
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