an abstract exercise.(lexicographic order)?

Question

let RxR be the lexicographic product of the group R, set of reals.
(a,b)<(c,d) iff a< c or a=c and b Show that any monotone sequence in RxR has a limit.
You can use that the same property holds in R.

well, in that case, it has limit plus infinity

yes euler i will think about it

Answer 1

Consider
{(0,1), (0,2), (0,3), (0,4), . . }
what does it converge to? (0,∞)?

Better yet:

What does {(0,0), (1/2,1), (3/4,2), (7/8,3), . . . , ((2^n-1)/(2^n),n), . . .} converge to? (1,-∞)? That's rediculous.

It's not true.

Answer 2

The same property doesn't hold in R.

1, 2, 3, ... is a monotone sequence in R which doesn't have a limit.
Are you missing a condition?