What's wrong with this bogus proof?

$\$0.01=(\sqrt{\$}0.1)^2$, not $(\$0.1)^2$.

You can clearly see the fallacy if you keep track of the units:

• In the second equality, $\$0.01 = \$0.1\times \$0.1$ is not true, if you are doing units.

• Even if the second equality were true, the third one gives problems: since $c=\$/100$, you have $$ (\$0.1)^2=\left(\frac c{100}\,0.1\right)^2=\frac{c^2}{100}\times\frac1{10}=\frac{c^2}{1000}. $$ This is not $(10c)^2=100c^2$.

In conclusion, two equalities are bogus, and so is the argument.