No strange at all. Actually there may be other similar threads. Lima has many many places to visit (depending on what you like). Historic center, beaches in summer, museums, bohemian and night life), LOT of good restaurants, etc.
So if you are not an outdoor traveler, hiker, etc. yes there are several places you’ll find interesting.
Those are the pros. Cons? Heavy, heavy traffic the whole day everywhere. Lima has become a nightmare talking about traffic. You must be patient and must plan your visits considering times and traffic.
You must go at least to Ica city if you want to take a touristic flight to Nasca lines. It’s 4 hours from Lima, and you may include a visit to Paracas nature reserve or Ballestas island (though I know you are not interested in places beyond cities, but I mention them anyway
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Now, a weeklong… I’m not sure. I guess 3 days in Lima would be more than enough. Why don’t you combine Lima and Arequipa? Arequipa is a beautiful city too, lot of architecture, churches, etc. For city tourism is a very good alternative. And it’s not so high, altitude shouldn’t be a big issue.