How do I prove that a boolean function constructed using $\vee$ and $\wedge$ (without using $\thicksim$ ) must attain the value 1 at least once?

Question

I made an attempt and got this solution. To prove this, lets construct a boolean expression using $\wedge$ and $\vee$ . In the boolean expression $(p\wedge q)\vee(p\vee q)$ , by entering the values $p=0$ and $q=0$ , we get $f(0,0)=1$ . Thus we have proved that a boolean function constructed using $\wedge$ and $\vee$ attains the value 1 atleast once. Is this correct? if wrong then how else do I prove it?