I'm in a real analysis class, and I love the material. But something that is coming up in homeworks worries me about my future math career, I find myself being tasked with proving things and using the internet to help me, finding the answer and completely understanding it, but that's not the problem. I have no clue where they got that from. It seems arbitrary when they pull some of this stuff out. For example:(adsbygoogle = window.adsbygoogle || []).push({});

I was told to prove the limit of a sequence is unique. I correctly started by assuming it isn't, which implies (by the definition of a sequence) x_{n}→L means ∀ε>0 ∃N∈ℕ such that n≥N ⇒ | x_{n}- L | < ε

x_{n}→M means ∀ε>0 ∃O∈ℕ such that n≥O ⇒ | x_{n}- M | < ε

The first thing they did was let n = max {N,O}. I understand why, the interesting stuff in the definitions happens when n ≥ N or O, so taking the max of these two ensures the interesting stuff is happening. But I don't think I could have come up with that myself. I'd like to believe i would, but I honestly don't know if I would. Since n is greater than or equal to N or O, both | x_{n}-L | < ε and | x_{n}-M | < ε are true, and I'm okay with this (it just comes from the definition), using the triangle inequality we can see that

| L - M | (which should be zero) ≤ | x_{n}- L | + | x_{n}- M | < 2ε

Now here's another part that messes me up, they chose [itex] \epsilon = \frac{| L - M |}{2} [/itex] at the very beginning of the proof! Which seemed arbitrary to me, so I ignored it until the end and this came up. I realized why they did that because it said | L - M | < | L - M |, which is a contradiction! Therefore the ε we chose has to equal zero, which is only possible if | L - M | = 0, therefore proving L = M. But how did they have such incredible foresight to see that [itex] \epsilon = \frac{| L - M |}{2} [/itex] at the very beginning? Just experience or is everyone in math a genius and I should just stick with physics (I'm dual majoring)?

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# How are things proven?

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