Revisiting liminf and limsup
This post is also available in PDF form from my WordPress Blog at https://explainingmaths.wordpress.com/2023/09/01/revisiting-liminf-and-limsup/ My recent series of posts about Fatou's Lemma for sums was aimed primarily at mathematics students at 3rd-year undergraduate level or above. So I think I should write a post more suitable for first-year undergraduate students. So I'm going to have another look at the topic of \liminf and \limsup for bounded sequences of real numbers (though most of what I say can be generalised to sequences, or even nets, of extended real numbers). I said quite a lot about this in a previous post on my WordPress blog in the post https://explainingmaths.wordpress.com/2009/01/10/an-application-of-absorption-to-teaching-lim-inf-and-lim-sup/ and there was a lot of discussion in the comments below the post. In particular, in those comments, I discussed the \liminf and the \limsup of some examples of sequences (x_n), including the following exam...