I think you missed the point.

So, firstly, you have split the particle 5 times. That's not infinite times. You can split it more, so that would be 6 times. And more. Even if you could split it 1000 times, that's not infinity.

The standard argument for infinity is that "you can always add 1 to any number, so there must be an infinity of them", and the refutation is that no matter how many times you add 1 to a number, all you've done is create a larger number. You never reach the point of actual infinity, no matter how long you keep doing this. You need to have infinite time in order to create an infinity by adding 1 to each number, so you're starting with the axiom that infinity exists (because you need an infinite number of operations to actually create an infinity). If you don't start with that axiom, then you can never reach infinity by addition (or any operation).