Chapter 5.3, Problem 45ES

In order for a proof by mathematical induction to be valid, the basis statement must be true for $n=a$ and the argument of the inductive step must be correct for every integer $k\ge a.$ In 45 and 46 find the mistakes in the “proofs” by mathematical induction.
45. “Theorem:” For any integer $n\ge 1,$ all the numbers in a set of n numbers are equal to each other.
“Proof (by mathematical induction): It is obviously true that all numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the inductive step, let $A=\left\{a_1,a_2,\mathrm{...},a_k,a_{k+1}\right\}$ be any set of $k+1$ numbers. Form two subsets each of size k:
$B=\left\{a_1,a_2,a_3,\mathrm{....},a_k\right\}$ and $C=\left\{a_1,a_3,a_4,\mathrm{....},a_{k+1}\right\}.$
(B consists of all the numbers in A except $a_{k+1},$ and C consists of all the numbers in A except $a_2.$ ) By inductive hypothesis, all the numbers in B equal $a_1$ and all the numbers in C equal $a_1$ (since both sets have only k numbers). But every number in A is in B or C, so all the numbers in A equal $a_1;$ hence all are equal to each other.”