Sequences and series: introductory exercises

Justify the existence of from the Archimedean Property and from Minimal Counter-Example. I.e. give a definition of what it means for = and prove for any real number there is exactly one such .

Solve the following inequalities, by finding the set of real values for which the inequality holds as a union of intervals. (Hint: divide into cases, depending on whether the argument of the absolute value function is positive or negative.)

(a) +1 1-

(b) ²⁺¹ 3

Solve the following inequalities, by finding the set of real values for which the inequality holds as a union of intervals. (Hint: be careful when multiplying out by checking that what you multiply by is positive or negative. In doubt, split into cases.)

(a) 5+1 +2 +13+2

(b) +1 +2 +2 +3