Deceiving citizens is typically considered the main political motive behind the spread of fake news. Accordingly, strategies to debunk fake news, such as fact-checking, have been suggested to combat it. However, the spread of fake news persists despite these debunking strategies. We propose an alternative but underexplored motive behind the spread of fake news: Fake news aims not only to deceive citizens but also to induce media skepticism. To support our claim, we present a stylized formal model of media skepticism and demonstrate that the incentive to spread fake news persists even if citizens are not deceived by disinformation coming from fake news. Our model highlights the dilemma embedded in fact-checking.