Is it possible to be reborn for all eternity? According to Buddhist teachings, is it possible to continue to reborn and suffer for all eternity?

Suppose if a being never attained any merit, never was able to work through its old karma. Would it be possible for that being to never escape suffering?

I have heard some people say:


  No, nobody can be reborn and suffer forever. For an extremely long time, maybe, but not forever.


I mean, forever is really an astounding concept. Even if you take billions of trillions of gazillions of births it's still nothing compared to forever.

If it's not possible to be reborn forever, then it implies that all beings are eventually destined for Enlightenment. Some may have an insanely long way to go, but if they're not gonna be reborn forever, then they have to eventually attain Enlightenment?

Do some beings continue to be reborn and suffer forever, or are all beings destined to attain Enlightenment eventually?

I prefer answers that give sources and all that. But some nice reasoning is also welcome.
Many people are interested in living forever (eg Christians) because such people do not think living is suffering. Instead, they wish to spend forever with their loved ones. That is why they are not queuing up to practise Buddhism. 

My impression is Buddhists are similar, in that they wish to believe in reincarnation so they can live more lives. That is why most Buddhists are not practising the path, hardcore, but preaching that Buddhists must believe in reincarnation, rebirth, whatever. 

Also, Buddhism does not state making merit ends suffering (SN 12.51) so what is its relevance?

Also, Buddhism does not state all beings are destined for enlightenment (AN 10.95).

Imo, the questions are an extreme &amp; pointless misrepresentation of Buddhism. If the questioner truly believe they were suffering, they would be a monk urgently putting out the suffering, like as if their hair was on fire.