Which of the following statements are always true and which are sometimes false? If the answer is sometimes false, give a counterexample; that is, an example of a sequence which demonstrates that the statement cannot always be true. If the answer is always true, indicate why by referencing a theorem or an idea in the book.

Explanation:

1.) If {an} > 0 for all n and limit approaches n of an = L, then L > 0 ALWAYS TRUE

2.) If {an} is bounded, then it converges

can be FALSE ; a_n = (-1)^n does not converge

3.) If {an} is decreasing, then it converges

can be FALSE ; a_n = -n does not converge

4.) If {an} is decreasing and {an} > 0 for all n, then it converges

TRUE

5.) If {an} is bounded, then {an/n} converges to 0

TRUE

6.) If [an} converges and {bn} converges then {an/bn} converges

can be FALSE; a_n = 1/n and b_n = (-1)^n/n

Both converge to 0 but a_n/b_n = (-1)^n which does not converge

the correct answer is ) possibility.

demonstrating opportunity cost is done through production possibility.

the opportunity cost can be demonstrated using the production possibility frontier or curve. this graphic shows the total number of possibilities for two products when we have the resources needed and used in an efficient manner. the main resources could be money, labor, and technology. the curve shows that the production of one good increases when the production of the othe

glass wants to live in a world of super heroes and villains. he have lived all his life trying to be responsible for the creating the origin story of a super hero; he schemed and caused many incidents (many of them fatal) hoping that someday someone with superpowers would show up and save the day.

he knew himself very much. he saw himself as a villain of a real-life super hero story. he was aware of his strengths (more than smart enough for being manipulative and scheming) and weaknesses (having fragile bones and having to rely on his mind to succeed).