I was reading a theorem giving conditions for a divergent series to have a convergent subseries and had a sort of flashback.

I studied nonlinear PDEs in grad school, which amounted to applied functional analysis. We were constantly proving or using theorems about sequences having convergent subsequences, often subsequences that converged in a very weak sense.

This seemed strange to me at first. If a sequence diverges, why is it of any interest that a subsequence converges? This seemed like blackout poetry, completely changing the meaning of a text by selecting various words. For example, here is the opening paragraph of Pride and Prejudice, blacked out to appear to be a real estate ad.

Here’s the big picture I was missing. We’re trying to show that a differential equation has a solution, and we’re doing that by some kind of successive approximation. Maybe our series of approximations doesn’t work in general, but that doesn’t matter. We’re just trying to find *something* that is a solution. Once you come up with a candidate solution, by whatever means, grasping at whatever straws you can grasp, you then prove that the candidate really is a solution, perhaps a solution in a weak sense. Then you show that this solution, potentially one of many, is unique. Then you show that your weak solution is a in fact a solution in a stronger sense.

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