In the ancient Hebrew book (http://dafyomireview.com/article.php?docid=398#ch5), Gate of the Unity of God, dealing with logical proof of creation of the world. The following premise is established:
PROOF OF SECOND PREMISE - (Beginnings are limited in number) The proof of the second premise is as follows: (commentaries to follow) Whatever has an end must have a beginning, because it is evident that something which has no beginning (i.e. existed eternally) has no end (i.e. is indestructible), since it is impossible for man to fathom the limits of that which is without beginning.
That's it. For some reason it is evident for the author of the book, but not so for me. Can someone shed a light why exactly something that has an end implies that it must have a beginning, and what is so difficult about imagining things that go infinitely into the past but stopped being in some well defined point in time?
UPD: Thanks lot for the answers. Thinking about it over the weekend I figured it out in the following way: Firstly, you have to consider the fact that the author of the book wasn't aware of Cantor and Hilbert so he probably had to apply a more simplistic approach to the infinity, which is more aligned with simple human understanding of the subject, and deals less with paradoxical mathematical puzzles on the modern mathematics.
It becomes really easy to understand when instead of the time continuum we move to the more material one. As in this example - "I have an infinite amount of money but then it ended" This does sound ridiculous. So is in Hilbert's Hotel- I had infinite number of rooms in the Hotel but then all of them became occupied. What would be the meaning of infinite in this case? So the same thinking applies to time continuum applying an "end" to the infinite sequence inevitably renders the sequence to be finite. Ask the following question: for how many times the sequence without a beginning (e.g. eternal) should exist? The answer would be is infinitely. But then how come it came to an end? This is the paradox the author is referring to.
The last note on one-sided sequence - like an example with negative number - the "catch" here is that we are tricking ourselves to think that negative number have started with some infinitely large negative number and then started to move toward the zero point, predating it in time. While the opposite is actually true for negative number to be negative zero point must be set firstly as beginning and not as ending point. So we're dealing with something that has a beginning and moving away from it infinitely, instead of something not having a beginning and moving toward its end.