Why can't the efficiency of a heat engine ever be $100 \%$?

(N/A) The efficiency of a heat engine is defined as $\eta = 1 - \frac{Q_2}{Q_1}$,where $Q_1$ is the heat absorbed from the source and $Q_2$ is the heat rejected to the sink.
According to the Second Law of Thermodynamics (Kelvin-Planck statement),it is impossible to construct a device that operates in a cycle and produces no effect other than the extraction of heat from a single reservoir and the performance of an equivalent amount of work.
For the efficiency to be $100 \%$,$Q_2$ must be $0$,meaning all heat absorbed from the source is converted into work.
This would require the sink temperature to be absolute zero $(0 \ K)$,which is unattainable according to the Third Law of Thermodynamics.
Therefore,some amount of heat must always be rejected to the sink,making $100 \%$ efficiency impossible.