About Analysis

if you're planning on taking real

analysis this semester or maybe you've

already started then I strongly

recommend watching this video until the

very end

so you can learn all the things I wish I

knew before I took real analysis because

no one ever really told me these things

before I got into it when you decide in

college that you want to be a math major

one of the very first quote real math

classes you take is real analysis it's

one of the first rigorous proof based

abstract awesome duda math classes that

you will take in your mathematics career

and these are the things that no one

ever told me the very first is that even

though real analysis is the same topics

as calculus you really you've studied

the same theorems the same ideas it is

totally different than calculus and this

is really the first big roadblock that

many students hit when they take their

first math major classes many math

majors they love calculus they take

calculus you know it's like high school

they do algebra and factoring in

derivatives and integrals and formulas

plugging things in it's almost like

solving a little puzzle and they love

that they decide to be a math major and

maybe they even take an introductory

proof class but really real analysis

seems to be like a roadblock just

because it feels so different even

though it is calculus I mean it really

is the same theorems it's totally

different you have to attack it with a

totally different mindset it's not just

formulas anymore it's not just plug and

chug anymore

it is a theoretic rigorous logic based

proof math class it's one of the big

three areas of mathematics and your

professor will treat it as such so the

first thing is just to get it out of

your mind you're not in the realm of

calculus anymore necessarily it's a

different level the second thing that I

really wish I knew was I wish I knew all

of the general proof techniques that you

would be using now

depending on where you take it depending

on where that lies in your degree you

may already know many of the proof

techniques or you may even be taught

them in class as you're going but when I

say proof techniques I'm talking about

things like mathematical induction you

know mutual inclusion if you're if

you're talking about when sets are are

equivalent

I just proof by cases you know all the

kind of general ways you prove things

depending on which real analysis class

you take these might be assumed

knowledge your professor might just

think hey we're just gonna start using

these proofs they know it already their

math majors of them you should know how

to prove things at least for me real

analysis was the second proof class I

ever took and some of it I knew the

proof techniques and many I didn't I was

very unfamiliar with just using

definitions to prove things in fact I

was just unfamiliar with how to write

mathematical proofs in general and just

kind of having a general idea of how

proofs work that's really gonna give you

a leg up it's something I really wish I

knew how to do at least a little bit

more going into my real analysis class

number three and this really goes along

with any math proof class you have to

understand how important definitions are

you sort of think definitions

definitions but when you become a math

major and you start doing math proofs

especially in real analysis it seems

like it's the first time this matters

more than anything is that you have to

know these definitions cold when I say a

sequence converges what does that mean

what does that really mean if a sequence

converges to a limit L you have to be

very specific and very exact because

these definitions show up in like every

single proof you do and if you don't

know definitions and you can't write the

exact definitions and you can't finish

the proof you can't solve a problem and

you're gonna get points taken off so

what happens a lot of times is students

may may prove it correctly may do the

problem sort of but they don't write it

in terms of

definition so what that means is they're

not really saying what the professor

wants them to say when your professor

writes a definition like alarm bells

should go off in your head this is

important

usually definitions are used to prove

things right proving things comes down

to knowing the definition I had an

instructor his joke was do you know the

definition of definition it's something

which you know by heart that was what a

definition was to him now the fourth

thing I wish I knew going into real

analysis is how to use definitions in

proofs many proofs especially real

analysis and math all over they work

essentially like this if you're given a

problem you usually have some

information you have some assumptions

and then there's something you'd like to

show based on those and so here's a very

nice way you can prove things you write

the definition of what you know and then

you might think you just attack the

problem and that's fine but one way to

do it is write what you know in terms of

its definition and write what you want

write the answer in terms of its

definition and then you try to sort of

just link these things up it's a really

nice way of doing things and actually

just writing the definition of the

assumptions and the conclusion that's

like half of the points almost right

there anyway all the points you're

missing is just the idea that links

these two things up and so if you kind

of like sandwich the problem you kind of

start at the end and the beginning it it

makes problems a lot easier if you don't

quite remember how to do them so that's

a really nice technique and I wish I

knew going into real analysis the fifth

thing I really wish I knew was how

important the logical quantifiers and

all of the math things all the math

terms were so the little symbols the

little symbols that look like this and

if you've seen that symbol so this is

like the is an element of so if I said

little a this symbol capital a this

means that little a is an element of the

set capital a so just just things like

that little notation

things become incredibly important in

classes like real analysis and depending

on which professor you have these might

be taken for granted

so my recommendation with this is really

to maybe take an introductory proof

class before real analysis rather than

jumping right in just so you can learn

all of these little notational things

these quantifiers as they're called and

also that introductory proofed

techniques which I discussed before and

the final thing the final thing I wish I

knew before taking real analysis is that

persistence is key the most important

thing especially with all these higher

level math classes if you want to do

well you have to persist you have to

stick with you got to you got to access

that gritty part of yourself because

these courses you know they're not

always easy they're very abstract and

the ideas are going to be new to you

they're gonna be new ideas and are going

to be hard to get your eye your head

around them and oh man I remember I

remember doing this and I remember

staring at the board and not

understanding it and it's like the

concepts just don't click and I would

practice and it doesn't click I'd

practice and it doesn't click and then I

practice and up finally starting to have

some breakthrough moments I personally

believe you don't have to agree with me

but I think like 90% of a math degree

it's just straight not giving up just

straight persisting and trying and

failing and not understanding and trying

again and then you finally get it your

brain finally adjusts and everything is

kind of cohesive so that's the last

thing I really wish I knew going into

real analysis is that you just have to

keep on going keep persisting keep

practicing and don't give up so those

are just the top things that were on my

mind today hope you enjoyed them and if

you're gonna take real analysis I wish

you the best of luck have a great day