Define a sequence of real numbers (xn) as follows: Let x₁ = 2, and supposing that an has been defined, define 1 2 3+1 = 2 ( x x + ²) Xn+1 . (a) Prove that x2 is always greater than or equal to 2, and then use this to prove that xnxn+1 ≥ 0. [So (n) is decreasing.] Conclude that lim = √2. in (b) For any real number c> 0, define a sequence (yn) so that (yn) converges to √c.

Prove that the function f : R → R is not continuous at xo = V2 f (x) = { if x E Q if x 4 Q x2 + 1 Explain why we can construct a sequence {xn}1 of rational numbers converging to v2. Have we proved this already?

Let n be a positive integer and let f : [0..n] → [0..n] be an injective function. Define the function g : [0..n] → Z as g(x) = n - (f(x))². Prove that is also injective.

A function f : Z → Z is defined by f(n) = 2n + 2. (a) Determine f(E), where E is the set of even integers. (b) Determine f-1(D), where D = {6k : k E Z}.