# Proof by induction for inequalities

## Homework Statement

A sequence (Xn) is defined by X1=3 and Xn+1= (6Xn+1)/(2Xn+5) for all n$$\in$$ N.

Prove by induction or otherwise that Xn-1 > 0 for all n $$\in$$ N.

## The Attempt at a Solution

I'm not sure with what to do when dealing with inequalities in an induction proof. Initial i tried subing in the recursion formula when attempting the inductive step but i dont think it gets me anywhere. I'd really appreciate any guidance on where to start.

Thanks

Related Calculus and Beyond Homework Help News on Phys.org
Mark44
Mentor
The proposition is clearly true for n = 1 and n = 2, so suppose it's true for n = k. I.e., that xk > 1.

For the induction step, you have to show that xk + 1 > 1.

xk + 1 = (6xk + 1)/(2xk + 5)

Carry out the division to get something that you can show is greater than 1. Is that enough of a start?

It is easier if you write it as x_n > 1 instead of x_n - 1 >0.

Suppose x_n > 1 [Show that x_(n+1) > 1]

Mult both sides by 4.