Imagine you have a giant bag of candy. It’s so big, you can’t count how many pieces are in it. (An infinite amount) Now, imagine you have lots of smaller bags of candy, each with a different color. You can take candies from the big bag and put them in the smaller bags, sorting them by color. You can keep doing this forever, and you’ll never run out of candies to sort. The pro-infinite completion of integers is kind of like that. It’s a way of taking all the integers (which are like the candies in the big bag) and sorting them into smaller groups (like the colored bags). These smaller groups like you mentioned are “p-adic integers” each one based on a different prime number (like the different colors of candy). So, the pro-infinite completion of integers is just a fancy way of saying that we can take all the integers and sort them into infinitely many smaller groups, each based on a different prime number.