Show that a sequence is monotonic Is there a standard way to show that a sequence is increasing/decreasing (monotonic)?
A sequence $a_n$ is (strictly) increasing if for all $n$, $a_{n+1} > a_n$.
For example, suppose we have the following sequence $$a_n = \frac{2n - 3}{3n + 4}$$ 
If my reasoning above is true, then I can simply show that $$\frac{2(n+1) - 3}{3(n+1) + 4} > \frac{2n - 3}{3n + 4}$$
and at the end we obtain $-4 > -21$, which is a true statement.
Does this proves that $a_n$ is increasing? Is this the general approach?
 A: That is one general approach: namely, ensuring that the difference of two consecutive elements is positive/negative.
Another, if the elements have the same sign (are all positive or all negative), is to ensure that the ratio of two consecutive elements is less than one / greater than one.
Yet another is to find a function on the positive reals that agrees with your elements at integral points and show that the derivative of the real function is everywhere positive/negative.
In other words, there is no single approach that is always used, just a toolbox of varying approaches.
A: If the sequence is recursively defined by $u=n=f(u_{n-1})$, one very general method is to prove the graph of $f$ is above the first bisecting line of the plane, i. e. if $f(x)>x$.
Naturally, very often the graph of $f$ is not entirely above or under this line on all its domain. The job consists then in determining on which intervals, if any, the graph is above or under the line, and to prove ther terms in the sequence all belong to one of those intervals.
