I don't understand a proof about volume and surface of revolution

Question

So I am investigating volume and surface of revolution, in particular the shape called Gabriel's Horn, which is $$\int_1^\infty \frac{1}{x}dx.$$

The interesting property about this shape is that it has finite volume but infinite surface area.The wikipedia article http://en.wikipedia.org/wiki/Gabriel%27s_Horn offers a proof that the converse is impossible, i.e there is no solid of revolution with an infinite volume and finite surface area. Here's the thing: I don't understand their proof! Sorry about the weird inequalities, that's how the proof was shown on wikipedia..

1. Let f be a continuous differentiable function in $[1,\infty)$

2. $\lim_{t \to \infty} \sup_{x \geq t} f(x)^2 ~-~ f(1)^2 = \limsup_{t \to \infty} \int_{1}^{t} (f(x)^2)' \,\mathrm{d}x$

3. $\leqslant \int_{1}^{\infty} |(f(x)^2)'| \,\mathrm{d}x = \int_{1}^{\infty} 2 f(x) |f'(x)| \,\mathrm{d}x$

4. $\leqslant \int_{1}^{\infty} 2 f(x) \sqrt{1 + f'(x)^2} \,\mathrm{d}x = {A \over \pi} < \infty.$

5. Therefore, there exists a $t_0$ such that the supremum $\sup\{f(x) \mid x \geq t_0\}$ is finite.

6. Hence, $M = \sup\{f(x) \mid x \geq 1\}$ must be finite since f is a continuous function, which implies that f is bounded on the interval $[1,\infty)$

I numbered them to make it easier to answer questions or to help me understand what is going on here...

I understand step 1 and 2, although I am unfamiliar with the concept of a supremum (basically the maximum x value right?)

I lose them from step 2-3, I don't know where the absolute value comes from. I also don't know wtf is going on in step 3, nor step 4.

I understand step 5 and 6.

Sorry for the long post

TL;DR: I'm a plebb and need help from steps 2-4 of this proof

Answer 1

$2\rightarrow3$:

This is just the inequality $\int g \le \int |g|$ for arbitrary integrable functions $g$. Furthermore chain rule is used $\frac{d}{dx}(g(x)^2)=2 g(x)\frac{d}{dx}g(x)$ .

$3\rightarrow4$:

Here we use the inequality $|g(x)|<\sqrt{1+g(x)^2}$ which should be obvious. Note also that in the last step the formula for the value of the surface of a rotational invariant surface is used $A=2 \pi \int dx g(x)\sqrt{1+g'(x)^2}$

Do you need more advice or can you take it from here?