Show that $f:= \frac{1 - 2x}{x^2 - 1}$ is monotonically increasing in the open interval $(-1, 1)$ Show that $f:= \frac{1 - 2x}{x^2 - 1}$ is monotonically increasing in the open interval $(-1, 1)$
To show a function is monotonically increasing, I started by saying that:

A function $f$ is monotonically increasing in an interval $(a, b)$ if for all $x, y \in (a, b)$, with $x$ < $y$, we have that $f(x) \leq f(y)$.

So I started by tryin to show:

Let $x$, $y$ be arbitrary fixed numbers in $(-1, 1)$, with $x < y$, we want to show that $f(x) \leq f(y)$:
  $$f(x) = \frac{1 - 2x}{x^2 - 1} \leq \frac{1 - 2y}{y^2 - 1} = f(y)$$
  We multiply the left side by $\frac{y^2 - 1}{y^2 - 1}$, which is positive because $y^2$ is always less than $1$, so we don't need to change the sign:
  $$\frac{1 - 2x}{x^2 - 1}\cdot \frac{y^2 - 1}{y^2 - 1} \leq \frac{1 - 2y}{y^2 - 1}$$
  We know multiply by $\frac{x^2 - 1}{x^2 - 1}$ the right side:
  $$\frac{1 - 2x}{x^2 - 1}\cdot \frac{y^2 - 1}{y^2 - 1} \leq \frac{1 - 2y}{y^2 - 1} \cdot \frac{x^2 - 1}{x^2 - 1}$$
  So we can simplify:
  $$(1 - 2x)\cdot (y^2 - 1) \leq (1 - 2y) \cdot (x^2 - 1)$$

Now, how do we proceed? Is this correct?

Without derivatives:
Note that we can write for $|x|<1$
$$f(x)=\frac{1-2x}{x^2-1}= \frac{-1/2}{x-1}+ \frac{-3/2}{x+1}$$
If we take $y>x$, then we have 
$$f(y)-f(x) =\frac{y-x}{2(x-1)(y-1)}+\frac{3(y-x)}{2(x+1)(y+1)}>0$$
for all $-1<x<y<1$
