the PDF of the $min(X_1,X_2)$

Question

So given two $r.v.$ $X_1, X_2$ who are $i.i.d.$ with probability dist. function:

$$p(k) = (1-r)r^k$$

with $k = 0,1,2,...$ and $0<r<1$

we want the Distribution of the $\min(X_1,X_2)$

So here is what I got:

$$P\bigg[\min(X_1,X_2)=k\bigg] = P\bigg[\min(X_1,X_2)\geq k\bigg] -P\bigg [\min(X_1,X_2)\geq k+1 \bigg] $$

$$P\bigg[\min(X_1,X_2)=k\bigg] =r^{2k} - r^{2k+2} = r^{2k}(1-r^2).$$

Is this correct?