Problem $1.(10$ marks$)$ Consider the following definition of Fibonacci numbers: F$($n$)=0$ if n$=01$ if n$=1$ F$($n$-1)+$ F$($n $2)$ if n $>22$ a$.(3$ marks$)$ Give the set of all natural numbers q for which F$($q$)>1\mathrm{.}69-2.$ Justify briefly. $($In the textbook, the set of natural numbers, denoted N$,$ starts from $0.$ This is what we adopt in this course.$)$ b$.(3$ marks$)$ Does your answer to a imply the following assertion: $3$c$,$ no $>0,$ such that $0<$ c$\mathrm{.}1\mathrm{.}6"$ no$.$ Explain briefly. c$.(4$ marks$)$ Consider the following recursive implementation of function F$($n$)($which differs slightly from the one given in the lecture notes$).$ procedure fib$($n$)$ if $($n $=0)$ then return $0$ else if $($n$=1$ or n$=2)$ then return $1$ else return fib$($n $1)+$ fib$($n $-2)$ Prove by induction that the procedure fib$($n$)$ is correct.