A price goes up and then down

Increasing $P$ by $a\%$ gives the new price

$$P_2=P+\frac{a}{100}P=\frac{100+a}{100}P.$$

Then the final price is

$$P_3=\frac{100-a}{100}P_2=\frac{(100-a)(100+a)}{100^2}P.$$

But

$$(100-a)(100+a)=100^2-a^2\le 100^2.$$

Therefore $P_3\le P$. Equality happens only when $a=0$; if $a>0$, the final price is strictly lower than the original price.