## 3.1. Sequences

### Example 3.1.3(a):

* =
{1, 1/2, 1/3, 1/4, ... }*,
which seems to indicate that the terms are getting closer and
closer to zero. According to the definition of convergence, we
need to show that no matter which
* > 0* one chooses,
the sequence will eventually become smaller than this number.
To be precise: take any
* > 0*. Then there
exists a positive integer *N* such that
*1 / N < *.
Therefore, for any *j > N* we have:

whenever| 1/j - 0 | = | 1/j | < 1/N <

*j > N*. But this is precisely the definition of the sequence

*{1/j}*converging to zero.

While it looks like this proof is easy, it is a good indication
for
'-arguments' that will
appear again and again. In most of those cases the proper choice
of *N* will make it appear as if the proof works like
magic.