How do epsilon delta proofs work in math?

Question

I have been trying to understand this part of math for a while, and no matter what it just seems very confusing.

In: Mathematics

Answer 1

If a number is less than the minimum of two real numbers then it is less than the two simultaneously.

Let ϵ>0ϵ>0

If δ=min(1,ϵ/4)δ=min(1,ϵ/4) then for all xx such |x−1|<δ|x−1|<δ we have

|x−1|<1|x−1|<1 and |x−1|<ϵ/4|x−1|<ϵ/4

However |x−1|<1|x−1|<1 implies that |x|−1<1|x|−1<1 (because |x|−|1|≤|x−1||x|−|1|≤|x−1|) so |x|<2|x|<2 and then |x+2|<4|x+2|<4 (because |x+2|≤|x|+|2||x+2|≤|x|+|2|)

Finally |(x2+x+1)−3|=|x−1||x+2|<ϵ/4×4|(x2+x+1)−3|=|x−1||x+2|<ϵ/4×4