If L is the limit of the sequence there exist such that . We don't know what the numbers , , ..., but there are only a finite number, even with L+ 1 added so there is a largest number in that set. That is what M is.

I'm not sure what yo0u mean by this. Obviously, . By the "triangle inequality", (equalityAlso, is the inequality with the equal sign combined technically equal in this case?mayapply but doesn't have to) and then is strictly less than because each such was between L-1 and L+1.

The "triangle inequality" applies to positive]As for the triangle inequality itself, does the less than part only or usually apply for the magnitudes of vectors or something?numbers. The lengths of the sides of a triangle are positive numbers, after all.

Any help would be greatly appreciated!

Thanks in advance!