Strictly speaking, no, it is not a requirement. But if you're not going to add more RAM, why bother upgrading your operating system? And your system won't crash, but it may run very slowly (because it is constantly paging memory out to disk).
In my experience with older machines, there are two things that will considerably help your enjoyment of your machine as you try to extend its life: 8GB of RAM and Windows installed on an SSD.