Trying to solve another induction proof problem (one part of a larger problem) and just trying to remember how to properly do this. Prompt and rough work below.

Question

mendicant_bias · Answer

$$\text{Given} \ x_{n+1} = \frac{1+x_n}{2}; \ \ \ x_1 > 1,$$$$\text{a.) Use Induction to prove} \ (\forall n \in N), \ \ \ x_n >1.$$

mendicant_bias · Answer

@ganeshie8

mendicant_bias · Answer

I never really got the hand of answering induction proofs that weren't built around a series but rather an inequality or equality, how do I do this?

mendicant_bias · Answer

I  guess I also don't understand on the inductive step how we know which equation to modify or use as the "assume x_k ..." relation. Which do we pick?

mendicant_bias · Answer

And I know the first step, it's just given in this problem to my understanding, x_1 > 1, right.

ganeshie8 · Answer

Hey still here ?

mendicant_bias · Answer

I'm here now.

ganeshie8 · Answer

Here is the complete proof : 

Base case : $x_1\gt 1$

Induction Hypothesis : Assume $x_{k} \gt 1$

Induction step : 
$x_{k+1} = \dfrac{1}{2}+\dfrac{1}{2}*x_k\gt \dfrac{1}{2}+\dfrac{1}{2}*1=1~~\blacksquare$

mendicant_bias · Answer

Ah, alright. Thank you!