(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Definition: Let a_{n}be a sequence of real numbers. Then a_{n}->a iff

for all ε>0, there exists an integer N such that n≥N => |a_{n}- a|<ε.

[for all of the following, "lim" means the limit as n->∞]

Theorem: Suppose lim a_{n}=a and lim b_{n}=b. Then lim (a_{n}+ b_{n}) = a + b.

Proof:

Given ε>0, find N1 s.t. |a_{n}-a|< ε/2 for all n≥N1 and find N2 s.t. |b_{n}-b|< ε/2 for all n≥N2.

Let N=max{N1,N2}. Then if n≥N,

|(a_{n}+b_{n}) - (a+b)| = |(a_{n}-a) + (b_{n}-b)| ≤ |a_{n}-a| + |b_{n}-b| ≤ ε/2 + ε/2 = ε.

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I am very confused about this proof and I'm never able to completely understand it since first year calculus.

In the definition, we have ε, but in this proof, WHY is it valid to take ε/2 instead of ε? What is the core of the reason that allows us to do this? I don't follow the logical flow of the argument. To me, taking it to be ε/2 instead of ε is like cheating...

2. Relevant equations

N/A

3. The attempt at a solution

N/A

I hope someone can explain this in more detail.

Thank you! :)

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# Homework Help: Rigorous proof of limits of sequences (2)

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