1. Solve and show using steps

a) Prove that + 1 4 1 for all real numbers 0

b) Find the smallest integer 0 such that 3 2 + 101 4 2 77 + 2 is true for all integers 0. Then, prove that 3 2 + 101 4 2 77 + 2 is true for all integers 0.

c) Find the smallest real constant such that 3 2 + 101 (2 2 19 + 20) for all integers 10.

d) A rational number is a real number that can be written in the form /, where and are both integers and 0. Given that 2 and 17 are irrational numbers, prove that the number 2 + 17 is irrational.

e) Prove that the number 5 0 + 5 1 + 5 2 is an irrational number. For this problem you cannot assume that any number is irrational to begin with. You MUST use the approach described in class (i.e., you cannot use prime factorization) and your solution MUST include a lemma demonstrating that if 2 is divisible by 5 then is divisible by 5.