Prove or disprove: If a canonical LP problem

Maximize: $f(x_1,x_2,$dots,$x_n)=c_1{x}_1+c_2{x}_2+$cdots$+c_n{x}_n$

subject $to{\displaystyle \underset{j=1}{\overset{n}{}}}{a}_{ij}{x}_j{b}_i(i=1,2,$dots,$m)$

$,x_1,x_2,$dots,$x_{n}0$

is unbounded, then there is a subscript $k$ so that the LP problem

Maximize: $f(x_1,x_2,$dots,$x_n)=x_k$

subject $to{\displaystyle \underset{j=1}{\overset{n}{}}}{a}_{ij}{x}_j{b}_i(i=1,2,$dots,$m)$

$,x_1,x_2,$dots,$x_{n}0$

is unbounded.