Consider the code below with computes $2$n for any integer n $>=0.$

function pow$($n$)$

k $=1$

c $=0$

while c $<$ n do

k $=2$ k

c $=$ c $+1$

end while

return k

end function

$($a$)$ Consider the following loop invariant for the while loop: At the beginning of

the while loop, we have k $=2$c

$.$ Prove that the invariant holds by induction