Discrete Mathematics Please use induction What is wrong with

Discrete Mathematics

Please use induction

What is wrong with the following proof of the (clearly nonsensical) claim that for every n epsilon N, 10^n = 1? Proof. We proceed by strong induction on n. Base case: n = 0. Then 10^n = 10^0 = 1. Inductive step: Suppose k epsilon N, and assume that for every natural number n lessthanorequalto k, 10^n = 1. Then 10^k + 1 = 10^2k/10^k - 1 = 10^k middot 10^k/10^k - 1 = 1 middot 1/1 by induction hypothesis. Therefore 10^k + 1 = 1, as required.

Solution

In this proof you used base case as n = 0,

n = 0 does not belongs to N

N = 1,2,3,4,......

So if we start with base case n = 1, the claim will be false in first step, as

when n = 1, 10^n = 1

10^1 is not equals to 1

OR

in the proof it should have been given that for every n >= 1, prove 10^n = 1.