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Although the idea of the proof is pretty simple, it involves some rather
fierce notation. Click here for the
proof in `R ^{2}`,
which
involves the key ideas, but without the need for complicated notation.

**Some Notation:**
In order to make the proof of the theorem go a little more smoothly we
need some notation.

If `x _{n}` is a sequence of points in the set

The
sequence provides a correspondence between the set of natural numbers and elements
of `X`, so the answer is that really the sequence is no more or less than a
*function*
from the natural numbers to `X`. In fact, in general, that's a any sequence is; a
function from the natural numbers into some set. From this point of view
it makes sense to write `x(n)` instead of `x _{n}`, or even to write;

So why do we generally use a different notation for sequences than for
all other kinds of functions? Just the force of tradition, together with
the fact that the notation does work well for most uses of sequences. In
the present situation it makes more sense to write our sequences as
functions `x(n)`.

What is a subsequence going to be in this new notation? A subsequence is
just a change of variable, or composition of functions, to get
`x(n(k))`, where `n(k)` is a function of the natural
numbers into themselves which must be
*strictly increasing*.

Now we'll use this notation to continue the proof of Theorem 6.5

We construct the appropriate subsequence in a series of steps. Click on each step in oder to proceed to the next: