Problem $2(5+5=10)$ Recursion and induction in binary codes

Digital transmission protocols transmit signals using binary codes. In order to minimize the effect of errors,

protocols often use codewords for signals with the property that $$similar$$ signals use $$similar$$ codewords.

One such code is a list of $2$n n$-$bit strings in which each string $($except the first$)$ differs from the previous

one in exactly one bit. Let us call such a list a bit$-$flip list since we go from one string to the next by just

flipping one bit.

Consider the following recursive algorithm for listing the n$-$bit strings of one such bit$-$flip list.

$$ If n $=1,$ the list is $0,1.$

$$ If n $>1,$ first take a bit$-$flip list of $($n $1)-$bit strings, and place a $0$ in front of each string. Then, take

a second copy of the same bit$-$flip list of $($n $1)-$bit strings, place a $1$ in front of each string, reverse

the order of the strings and place it after the first list.

For example, for n $=2,$ the list is $00,01,11,10,$ and for n $=3,$ we get $000,001,011,010,110,111,101,100.$

Prove the following two statements by induction on n$.$

$($a$)$ Every n$-$bit string appears exactly once in the list generated by the algorithm.

$($b$)$ Each string $($except the first$)$ differs from the previous one in exactly one bit.