Suppose A be m-by-n matrix

Suppose A be m-by-n matrix

Show that if $m>n$, then rank(A)<$m$ , and that if $m<n$, then nullity(A)>$0$

My idea:

If m>n , suppose Rank(A)>m then rank nullity theorem rank(A)+nullity(A)>n

how to prove next one

Hint:

if $m < n$, $\operatorname{rank}(A) \leq m$

$$\operatorname{rank}(A) + \operatorname{nullity}(A)= n$$

$$\operatorname{nullity}(A) = n- \operatorname{rank}(A)$$

Try to obtain a lower bound for $\operatorname{nullity}(A)$, from what I have written.