I already stated that I'm not a Buddhist and I don't believe in the Buddhist teaching of rebirth. I am very interested in Buddhism, however.

it doesn't follow that you have practised since beginningless times and you have already practised with diligence infinite times and you somehow always failed. — boundless

If a cycle of rebirth and death is beginingless then there will always be a previous cause or rebirth and this would go back infinitely. If there’s no beginning then there’s no end.

This isn't necessarily the case. Traditional buddhists would reply that the ultimate cause of the cycle is ignorance. If ignorance is removed, samsara stops. If ignorance is never removed, the cycle will go on forever.

BTW, this problem was one of the reason why I ultimately ceased to try to become a Buddhist. If the cycle is beginningless, then the very existence of the 'cycle' is unintelligible. As youb remarked, each instance of rebirth is intelligible in principle, it is a regulate phenomenon. It would be weird if the very existence of the cycle is an unintelligible 'brute fact'.

If, however, the cycle began, this means that if other traditional Buddhist claims are true it must end:

“Whatever is subject to origination is all subject to cessation.” — SN 56.11, bhikkhu Bodhi translation

However the traditional Buddhist view is that it doesn't necessarily end. Rather it ends if ignorance is removed.

If the cycle is beginningless, then the very existence of the 'cycle' is unintelligible. — boundless

'Saṃsāra has no beginning, but it has an end. Nirvāṇa has a beginning, but it has no end' ~ Buddhist Aphorism (quoted on Dharmawheel.)

This isn't necessarily the case. Traditional buddhists would reply that the ultimate cause of the cycle is ignorance. If ignorance is removed, samsara stops. If ignorance is never removed, the cycle will go on forever. — boundless

But we’re not looking forward, we’re looking infinitely backwards, and in the past ignorance has necessarily never been removed because we are here in ignorance.