**To prove that e is irrational**

We define

_{}

Now each term of *a _{n}* is
positive, so

The terms of *b _{n} *are a geometric series, so

_{}

Moreover in comparing each
term of *a _{n}* with the
corresponding term of

_{}

etc, so it follows that *a _{n}* <

Hence we have
_{}.

By definition,

_{}.

Now suppose *e* is rational, ie that

_{}

Then we have

_{}

We can choose *n *so that *q *divides *n* (eg *n* = *q*).

Then the expression enclosed
by inequality signs is an integer.

An integer cannot lie
between 0 and _{}.

That is a contradiction, so
the assumption (that *e* is rational)
can only be false.

Hence finally ** e
is irrational**.

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