Yes, right in Stockholm city there is skiing! (if there is snow) Not exactly the alps but a place where you can get your daily dose of needed skiing.
Hammarbybacken is the name, at the end of the Tvärbanan in Hammarby Sjöstad, next to Södermalm. Check the current snow depth on their webpage
Late January, beginning of February Hammarbybacken in Stockholm city will hold the Alpine World Cup in downhill. The premiere last year was a success and it seems to be a cold and snowy winter suitable for city winter sports this year. The world’s best skiers are gathering in Hammarbybacken to compete for podium places in the Alpine World Cup parallel slalom competition for men and women. Parallel slalom races are run by night – really fun events with high tempo. Quite spectacular, having ski races in the Stockholm city center.