Real analysis is a joke. There are too many unintuitive tricks required that you "just have to know". What kind of normal person "adds zero" to an equation by adding and subtracting the same thing, this trick isn't pulled anywhere else regularly but it's the kind of shit you pull all the time in analysis.

Beware Cat Shirt $21.68 |

Beware Cat Shirt $21.68 |

more like fake and gay analysis amirite

First of all I fricking hate the title of that theorem, it's split up into directions anyways a normal person would write: cauchy sequences are convergent then a second theorem covergent sequences are cauchy.

Anyways, this trick is just partitioning, it's actually quite common especially in analysis especially in cases where triangle inequality is optimal.

We need an a_n_k so we break our distanced into two pieces.

You need to improve your geometric reasoning.

it's only bad because they don't explain it. You can define 0 as:

for any number a there is a number -a such that |-a| = a therefore

a + (-a) = 0

and 1 as

for any non zero number a there is a number 1/a such that (1/a)^-1 = a therefore

a * 1/a = 1

instead of treating 0 and 1 as representation of a quantity you see them as "no net change", and that is useful in analysis because whole calculus is based on dividing by dx when lim dx->0 which you have to somehow overcome by mixing equations around. For whatever reason no one presents this as a valid technique to be kept in your mind when solving all analysis related problems

> what do I lack to make this expression look better?

instead it comes at you out of nowhere making it seem as if someone just randomly found what works in this specific case

delta/epsilon arguments drive me mad. Like "let's choose delta/2 in this example just because...reasons". I'm convinced people teaching analysis don't generally understand what is going on, they're just reciting book-memorized proofs line by line "because it just works like that" and more than any other field there is a ton of hand waving and "this is obvious" or "I won't bother to do show the reverse "...and only if" direction of the theorem because it's trivial and you can do it yourself as an exercise". Then when the answer comes it's some completely different idea involving some weird inequality or trick you could never dream up yourself without 20 years experience. I hate analysis with a passion.

yes, there is a big need for "let's show how one would think about it' rather than just showing what work without context

> b-but my examerino that student have to pass in 2 months...

that's why analysis is bad, like all classes

Yea they can be annoying since its often not obvious what epsilon to choose. The best strategy is to wait until the end of your proof to see what you need to make it all work.

I always liked delta-epsilon language and devised many proofs with it. coming up with the correct delta value feels like solving a chess puzzle. for plebs there's a mindless way, just pick delta for delta, you'll get some convoluted formula where epsilon should be, make it equal to epsilon and solve for delta.

>Like "let's choose delta/2 in this example just because...reasons".

this is exactly what it is

and by learning more proofs you get better at picking the magic values to work backwards from

Are you single?

Why are you going around sci asking people if they're single

assessing the possibilities of finding a smart anon who is single.

important facts about me:

> not single

> bio major

When analysis is not taught humanely, you can be overwhelmed by a lot of notation for not saying much.

Given a convergent sequence in a normed vector space, one can show that the sequence of partial arithmetic means goes to the same limit in one sentence, behold.

Given arbitrary positive epsilon, there is a big N so that for small n greater than big N, the small nth element of the sequence is within epsilon over two of the given limit, L, and a straightforward calculation that is beneath me shows that the distance from the small nth partial mean to L is bounded by a weighted average of the maximum distance of the elements of the given sequence to L and epsilon over two, the weight of epsilon over two going to one.

If I had to grade anything longer than this, I would not be happy.

>too many unintuitive tricks required

name another one.

If you are being filtered by the triangle inequality then I feel sorry for you.

If-then clauses take a comma.

I'm sorry the meaning was completely lost by not having a comma.

there's tons. You do weird shit like randomly declaring that (a+b) is less than or equal to 2*max(a,b) and then go off on some thread with that idea. Or something with the binomial theorem where you only use one term and throw away the rest because "trust me"

>(a+b) is less than or equal to 2*max(a,b)

This is just basic inequality stuff. If it seems random then you better reverse engineer some kind of pattern/explanation/heuristic to help you guess when to use it.

>Or something with the binomial theorem where you only use one term and throw away the rest because "trust me"

Ignoring terms when shit gets big or small is what calculus is all about.

Wilderberger was right about analysis.

Infinite sums in general is where it all starts to become "just trust me" and the intuition just gets harder and harder to follow

Just wait until this homosexual learns about the cantor set

Or summing divergent series...When you get there the rules don't matter anymore.

You're just low IQ. Analysis is based on geometric intuition. Splitting a distance into two segments with an intermediate point is perfectly intuitive.

I sort of get it, the level of geometric reasoning at that point is large but before that it was almost nonexistent, especially if you're American since education is so bad.

Personally I just dealt with it since I already had good spatial reasoning, so I don't know about any specific books but I'm sure there are some primers out there that can help improve OP's thinking about mathematics.

>"adds zero" to an equation by adding and subtracting the same thing

literally everything does that. x-1=0 -> x=1 another example