Another way to see this is that a computer with limited memory can be modeled as a finite state machine. For every possible arrangement of memory and other inte

Another way to see this is that a computer with limited memory can be modeled as a finite state machine. For every possible arrangement of memory and other internal state, you have a node. Include the finite input as part of the state too. If your computer is deterministic, every state leads inevitably to one next state. If your computation runs infinitely and there are finite states it can hit, it must hit the same state twice. Since there's only one thing it can do from that state, it will just repeat the same thing it did on its first visit to the state and eventually come back. So the machine is in an infinite loop. This is true even if you remove the output storage from your model. You could have external storage and label each state transition with what it writes to the output. This tells us that either the output is finite or it's periodic. The digits in an irrational number are not periodic, so a finite machine can only output all the digits in a rational number.