# Famous Limit

is a famous limit that is in all Calc I books. Here are some proofs about it that are usually done other ways.

**Theorem IA**

- is increasing

Proof:

A method **Polytope** saw someplace.

Consider the numbers

These have a geometric mean.

These have an arithmetic mean.

Now we use the Arithmetic Mean-Geometric Mean Inequality (*http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means*).
and

**Theorem IB**

- is decreasing.

The proof of **IA** can be adapted to prove **IB**.

Consider the numbers

These have a geometric mean.

Next we can compare to the harmonic mean, , using the Geometric Mean - Harmonic mean inequality. (*http://planetmath.org/encyclopedia/ArithmeticGeometricMeansInequality.html*)

**Theorem IC**

The sequences from **IA** and **IB** both converge to the same limit.

**Proof**

Since the sequence of **IB** is decreasing and positive it is bounded below by **0**. The sequence of **IB** converges due to Proposition 3.1.9 at this site. (*http://pirate.shu.edu/~wachsmut/ira/numseq/sequence.html*)

For all n,

So the convergence of the sequence from **IA** follows by the same proposition.

So both sequences have the same limit.

**Lemma IIL**

**Proof**

As increases, the integral smoothly and monotonically increases beause its first derivative is . Suppose it reaches a finite limit . Use the substitution .

Since cannot be ,

**Theorem II**

Let

Since is monotonically decreasing, it maximum occurs at 1 and its minimum occurs at . We can use these and the difference of the limits, , to bound the integral.

Using the squeeze theorem (*http://en.wikipedia.org/wiki/Squeeze_theorem*). Then from the **Lemma IIL** is finite since the limit of the integral is infinite for an infinite upper limit.