# Foundations and Analysis I – Revision Steps and Tips

For all those cramming in some final revision for their Foundations and Analysis I exams this holiday, I hope this guide helps!

**STEP 1: Definitions**

**Always** start with definitions! Okay, so this is probably the only logical place to start… but, sometimes it’s easy to underestimate the importance of definitions. Knowing your definitions perfectly, is not only the best way to form a solid foundation for the rest of the module contents, but, in itself puts you in a very good position for the whole exam.

Definitions may not be the most exciting, but they are almost definitely the easiest part to learn – don’t take them for granted!

**TIP**: make sure you can easily adapt definitions to define their converses – if you can define ‘convergent’, you are expected to also be able to define ‘not convergent’!

**STEP 2: Theorems**

Especially the ones with names!

**TIP**: it’s much easier for the examiner to ask for a definition of the “Well Ordering Principle”, than Proposition 3.12… – so, as a general rule of thumb, prioritise theorems (propositions/ lemmas/corollaries!) with names.

**STEP 3: Proofs**

Learn as many proofs as you can! Focus on the key steps, or that one trick step. Summarise proofs in these key steps or even an explanatory diagram and learn these – you should be able to fill in the rest on the day!

**TIP**: when you’re stuck on a proof, write down all the relevant definitions. Definitions can often help you spot the next step, and if not might pick up a few marks in themselves – told you your definitions would come in handy!

**TIP**: You don’t always have to write the proof from scratch. Make sure you know the approximate order of the material – you’re generally allowed to assume any pre-proven material. Use the number of marks a question is worth as a guiding factor too!

**STEP 4: Practice, Practice, Practice!!!**

Past papers are your best friends!

It can always be daunting going for your first past paper, especially if you’re a bit of a perfectionist! But, dive in – and see what they’re like… and then see what the others are like – look familiar? If the definition for convergent is the first question of the last three Analysis I papers, although I can make no promises, I’ll let you guess what the first question of this year’s paper is going to be!

Have a look through past papers for favourite definitions, topics and proofs and make sure you’re good at them, because it seems some questions stand the test of time!

**TIP**: Look for the past papers written by the lecturer that teaches you now – every lecturer has their own style and favourite questions!

**TIP**: Step 4 is the most important step! Whether you’ve thoroughly done all three of the previous steps or are feeling a little bit anxious just thinking of your effort on step 2, if you’re running out of time go straight to step 4 – highlighting lecture notes and making flashcards just isn’t quite the same as trying out the real thing!

Best of luck with your revision!