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Homework Statement Prove that if f is continuous at a, then for any ε>0 there is a σ>0,? such that if abs(x-a)< σ and abs(y-a)< σ then abs[f(x) - f(y)]< ε Homework Equations Definition of continuity and triangle inequality abs(f(x)-f(y))= abs(f(x)-f(a) + f(a)-f(y))≤ abs(f(x)-f(a))+...

that was sort of my attempt at a solution. This is what I've got now. Assume a irrational, but √(1+a) is rational. Then √(1+a) = p/q for p,q integers q≠0. 1+a = p^2/q^2 → a = p^2/q^2 - 1 → (p^2- q^2)/q^2 Since p,q integers, p^2-q^2 and q^2 must be integers. Thus a must also be rational by...

Homework Statement Suppose a is irrational, prove√(1+a) is irrational. Homework Equations A number is rational if it can be expressed as p/q, p,q integers with q≠0 The Attempt at a Solution I can reason through it intuitively but not sure how to demonstrate it formally. Any...