1. Page 10, 1.2.6*. Show that the following sequences of sets, {Ck), are non-decreasing or non-increasing give definition of limit k→ Ck and then find it in each of the following cases. (a) Ck={x: 1k≤x≤10-(1k)} k= 1,2, ... and for (b) Ck={(x,y):1/ksx2+y2≤2-(1/k)}, k = 1, 2, 3, ... (c) Ck={x:0

2. A. Indicate whether the following functions are: 1-1, onto, total by circling all valid options. The domain and codomain are specified. Justify briefly. (i) f(x)=x. Domainis N.Codomainis N. 1-1 Onto Total (ii) f(x)=x2.Domainis N.Codomainis N. 1-1 Onto Total (iii) f(x)= x+1. Domain is Z+. Codomain is Z+. 1-1 Onto Total (iv) f(x)= [x/2]. Domain is N. Codomain is N. 1-1 Onto Total

1. Let f: N→ N be a rule defined by n f(n) = 2 if n > 1 2n + 1 ifn ≤ 1 (a) Is f a well-defined function? Justify your answer. *Well-defined function: the function applies to all in the domain, outputs are all in codomain, inputs have one single output. (b) What is the domain and codomain of this function? Justify your answer. *Domain - set of all inputs for a function/ Codomain - set of all possible outputs/ Range - outputs of f for some input x (c) Is this function injective, surjective, or bijective? Justify your answer.

6. Given that -뿔 (금) (규 R(x) 4! 1+. for x E (-, 3), where & is between x and 0, find an upper bound for |R|, valid for all x € [-,), that is independent of r and §. 21 2