Understanding how to prove limit theorems for sequences. How do you know what arbitrary value to choose for epsilon as in this document, and when should the triangle inequality be considered when writing your proof? 
 A: The Triangle Inequality will be a tool you use very often in analysis. Many definitions in analysis end with $``<\epsilon$'', so the Triangle Inequality is used when you have determined that certain terms have bounds and you note that another expression you want to bound can somehow be written in terms of those earlier terms you have a bound for already.
For example, in the document you linked, in Theorem $311$, they know to use the Triangle Inequality, because they have established the bounds on the terms on the right hand side of the Triangle Inequality, namely $|a_n-L_1|<\frac{\epsilon}{2}$ and $|a_n-L_2|<\frac{\epsilon}{2}$, and so the Triangle Inequality allows you to bound $|L_1-L_2|$, since it allows you to note that $|L_1-L_2|=|L_1-a_n+a_n-L_2|\leq |L_1-a_n|+|a_n-L_2|<\epsilon$.
A: First of all, I don't know what you mean by "tricks one page 1". If you mean this: These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added, multiplied, divided, ... Theorems from this category deal with the ways sequences can be combined and how the limit of the result can be obtained. If a sequence can be written as the combination of several "simpler" sequences, the idea is that it should be easier to find the limit of the "simpler" sequences. These theorems allow us to write a limit in terms of easier limits. however, we still have limits to evaluate. The second category of theorems deal with specific sequences and techniques applied to them. Usually, computing the limit of a sequence involves using theorems from both categories. -these aren't really tricks so much as an intuitive description of what the theorems mean and how they're used. 
The first is a general strategy for attacking mathematical problems in general, a divide and conquer strategy. The classic examples is the method of exhaustion for finding areas and volumes developed by the Ancient Greeks, in which a general region or object is subdivided into polygons,spheres or circles and the areas or volumes of the individual components are added up. This method, of course, eventually evolved into modern theories of integration, which are much more precise, but the general notion of subdividing the unknown into known quantities is still the main idea. 
The use of the triangle inequality is a major component of this approach-and a general tool in analysis period-because it states that the absolute value of a sum is less then or equal to the sum of the individual absolute values,which are usually easier to computer or approximate. The triangle inequality-in it's various forms-is used when we need to rewrite an absolute value in parts which we know the $\delta-\epsilon$ bounds of and we can then add them to give the bound for the entire expression. A perfect example is the proof that the limit of a sum is the sum of the limits for sequences in Theorem 3.11. Study it carefully. 
This is part of a more general problem beginning students have with careful calculus. Analysis is the study of inequalities, of estimating quantities within certain bounds. This is why most American students coming out of pencil-pushing calculus-where everything is equalities and algebra of absolute quantities that can calculated mindlessly-are completely shell shocked by their first analysis course, where everything needs to be proven by estimation. 
My suggestion to you is to study as many computed examples as possible, as well as as many inequalities as you can. Inequalities are as basic to analysis as the trigonometric identities are to basic calculus and you need to get comfortable with using them.    
