This is quite challenging to prove from scratch. Do you have some tools you've learned that you might use?
– Eric WofseyNov 28 '18 at 21:42
1
You can find a full proof (not just a hint) at https://math.stackexchange.com/questions/320444/why-isnt-an-infinite-direct-product-of-copies-of-bbb-z-a-free-module.
– Eric WofseyNov 28 '18 at 21:45