Real Analysis Feedback

As it turns out, that wasn't as bad as I thought it would be. Had some significant time constraints, and some of the concepts of continuity really threw me near the end but it wasn't a total nightmare.

Must admit though, there are still some areas which I feel I still need to "grok". (Hmm... never much liked that word... but replacing it with "understand intuitively" doesn't quite sound right either, but you know what I mean.) Concepts of continuity is one such area that I'll have to spend some extra time on in the revision stage (or maybe even just before my final Analysis section).

In summary, this was certainly the most challenging section yet. Having said that, I did manage to achieve a higher mark than I expected to get in my assignment. As usual, here are some areas in which I screwed up:

It seems intuitive to say that  $0 + \infty + 0 = \infty$, but this specific rule regarding the sum of these limits was never listed in the set that I can use, so I can't use it, hence I was marked down. I've always enjoyed working with a limited toolset, so this should come naturally after a bit of revision. The main problem is getting that full understanding of the mechanics of each rule so it can become intuitive. In fact, my answer for this particular question (as a consequence) was extremely drawn-out. The proper answer given by my tutor fits on less than half a page of A4. Hopefully more practise will let me see the quick, correct answer more quickly.

I stated Bernoulli's Inequality incorrectly. Absolutely no excuse for that. 🙁

When working out limits of formulas, always state the dominant term before reducing, and always put curly brackets around a sequence (otherwise, it's just a formula).

In some cases I was lazy and claimed that something like $\frac{3}{n+2}$ was a basic null sequence. Although it is quite obviously basic, and quite obviously a null sequence (it converges to 0 as n increases), it's not actually a basic null sequence. So even something as small as this, I need to deconstruct and prove.

Lastly, and this does bear repeating... I need a lot more practise with questions about continuity...

Next up, more group theory!

Time Constraints

I'm coming towards the end of my last section of my Analysis. I'm probably about half-way through the last book. After that, I have to answer the last question in my Analysis assignment, proof read the whole thing and submit it.

I have the whole of today, Monday, Tuesday and Wednesday before I need to submit this Analysis assignment. To make sure I can submit this assignment in time, and get ahead for my next section (Group Theory Part 2), I've taken the Monday, Tuesday and Wednesday as holiday days. That'll make a total of 5 days I've taken off work since the beginning of January just to make sure I'm where I want to be with my study.

There's a ebb and flow to where I am with my study compared to where I should be, and I won't truly understand how well I managed my time until this whole module is over in June, but I sure wasn't expecting I'd have to use holiday days to catch up on study.

One of the most frustrating things about feeling like you're behind is the nagging feeling like you don't have sufficient time to learn the materials sufficiently. This slight panic creeps in and you realise that more than anything else, you need to reach the end of the section (to be able to move on to the related assignment question). So you learn it JUST well enough to move on. Of ALL the sections for this to happen on, it had to happen with Analysis didn't it. The section I was most nervous about.

So in summary, because I've found myself short on time, I'm having to practically rush through the section on Analysis. No time for playing with concepts, no time to study the proofs in depth (or at all in some cases). Just time to get the general gist, and move on.

Something I will say is that I haven't found it necessary to keep a spider diagram of how core concepts relate to one another. Because of the structure of the learning materials I find it fairly easy to see the links and how one theory supports another proof and so on.

What has been a surprise while studying analysis is that there hasn't been a need to write any proofs in quite the same way as I was expecting. It turns out that (so far, at least) there isn't a need to have an in-depth understanding of logical notation or concepts. You need a basic understanding of the "if x, then y" structure, and the consequential converse "If not y, then not x", and so on, but not a great deal more. I'm not sure whether I'm grateful for this or not. Some of the concepts in Analysis require all my brain power, so I'm not sure the additional logical puzzles on top of that would help. Though on the other hand, having a decent foundation in logic feels quite important. I suppose, again, I'll have formed a more solid opinion on this by the end of the module.

Linear Algebra Feedback

Received another good mark for this assignment, which I'm happy about as at the last minute I realised I'd messed one question up so re-did it. Though despite the high mark, I had a lot of great feedback from my tutor.

Overwhelmingly, he mentioned that I had to use more English to explain why I was solving a question in a certain way. This struck a chord with me as the more solutions I read, the less English there appears to be explaining the methods used, and ever since starting this Mathematics journey this is something I've had issue with.  Some people find Mathematics hard to get into because of the certain barriers to entry, ie: the text won't fully explain what a variable is or (case in point) won't explain how to get from one step to the next. Often such a step is taken as "obvious", and I've always considered this as somewhat "elitist". With this assignment I've face-palmingly found myself falling into the same trap.

This is doubly interesting for me, as I actually do really enjoy going into lots of detail in explaining something I've worked on. So in future, I'll be mindful to explain more of my workings in plain English; or at least summarise them in plain English.