Can we approximate a function $f\in L^1([0,1])$ by a function $g\in L^2([0,1])$ in $L^2$ norm?

I am solving a problem and the problem asked me to approximate a function $f$ to $g$ in $L^2$ norm as stated above. It seems not true since there is no condition like $L^1\subset L^2$ or $L^2\subset L^1$...... neither space cannot be dense in another space... I have no idea... I will be thanking for any hint or answer!