2 Convergence of sequences

Having defined real numbers, the first thing we will study is sequences. We

will want to study what it means for a sequence to converge. Intuitively, we

would like to say that 1

,

1

2

,

1

3

,

1

4

, ···

converges to 0, while 1

, ···

diverges.

However, the actual formal definition of convergence is rather hard to get right,

and historically there have been failed attempts that produced spurious results.