I’ve recently finished teaching 5 weeks of Analysis as part of our new first-year Core Mathematics module. I’ve made some additional resources available to them on a Moodle page. I expect to add further resources in due course! But here is the current version of that page.
Optional additional resources
Dr Feinstein’s blogs
I have two blogs about explaining mathematics. My WordPress blog at https://explainingmaths.wordpress.com/ has been running for a long time, but the version of LaTeX available there does not look as good as the more modern MathJax LaTeX available elsewhere these days. So I now also have a Blogger blog at https://explaining-maths.blogspot.com/
My two blogs do link to each other. Currently I write most of the mathematical posts on https://explaining-maths.blogspot.com/ and then post a link to those posts on https://explainingmaths.wordpress.com/
Taylor’s Theorem
This Blog post includes a two-function version of Taylor’s Theorem (similar to the Cauchy Mean Value Theorem, CMVT). This is something I thought about while teaching the material on differentiation in Analysis. It gives an alternative route in to proving Taylor’s Theorem via what I called a Hybrid Lemma, which resembles a combination of CMVT and L’Hopital’s rule. This two-function Taylor Theorem doesn’t appear to be widely known, and I suspect its main application is to prove the usual version of Taylor’s Theorem with (Lagrange form) remainder.
Mathematical Analysis
Another old module of mine, G12MAN Mathematical Analysis, is available in full for viewing or downloading from the University of Nottingham’s Open Educational Repository UNoW (http://unow.nottingham.ac.uk/). There is also a YouTube PlayList at https://www.youtube.com/playlist?list=PL58984C080F2B0575
Warning! In an attempt to help students understand some of the concepts better, I introduced some non-standard definitions: absorption of a sequence by a set; and the set of non-interior points in a set \(E\), denoted by \(\textrm{nint}\,E\,.\) Although potentially helpful for at least some students, these concepts have not caught on widely (yet!). So you should not use the terms absorption or \(\textrm{nint}\) in your work unless you include an explanation of what they mean. Your lecturers are unlikely to have heard of these terms!
Although much of the material concerns the topology of finite-dimensional Euclidean space, there is also some overlap with the analysis you are learning now. In particular, the chapters there on differentiation and on integration might be interesting (although the material on the Riemann integral is very compressed, and most of the results there are not proven). There is also a chapter on sequences of functions, where you can find out more about pointwise convergence and uniform convergence.
Introduction to the Weierstrass M-test
I am writing a series of posts on my Blog on the Weierstrass M-test.
Core Topics in University Mathematics
Oldies but goldies! This playlist on YouTube includes 20 videos on topics that students often find challenging in first year. You’ll find some videos by me on Think of a function and on Sequences and their properties.
Foundations of Pure Mathematics
Here you can find some links to full sets of recordings of classes from various editions of a previous module of mine, Foundations of Pure Mathematics (FPM), which used to be the introductory module on pure mathematics in autumn semester of first year here. In particular, the recordings from the 2014 edition of FPM are available on YouTube Obviously things have changed a bit, but if you do want an alternative look at some of the topics, you are welcome to have a look!
with the usual dot product (and also 
when regarded as a two-dimensional real inner product space)17/4/25Some musings on Cauchy-Schwarz, Part IVbIntroductory discussion of the case of two-dimensional real inner product spaces, and a summary of what we have already said about them. Also some discussion of why I think it might possibly be worth spending all this time on such a well-known inequality, when there are short and elegant proofs available of the general case.16/4/25Some musings on Cauchy-Schwarz, Part IVaThe trivial case case of one-dimensional inner product spaces over 
8/4/25Some musings on Cauchy-Schwarz: Part IIConnections between real and complex inner product spaces, and especially between 
