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Anyways first proof:

Im starting off with something easy because right now I am more focused on concentraiting on the correct syntax and verbage and what not that I should use.

I want to show that f(x) → a^2 near a.

Proof:

For every ε>0 there is a δ>0 such that for all x, excluding x = a, if |x-a|< δ then

|f(x) - a^2|< ε

I am starting my proof off with a definition which states important facts namely ε and δ must be greater than zero for ALL x and at the same time explaining what must be true if we expect the limit to exist. In general is this how proofs are normally started by briefly explaining what must be true?

|f(x) - a^2| < ε = |x^2 - a^2| < ε = |x+a||x-a| < ε.

For something as easy as this, is it necessary to go step by step from simplifying |f(x) - a^2| to |x+a||x-a|?

|x| - |a| ≤|x-a|< δ1, |x|<δ1 + |a|, |x+a|≤|x| + |a|< δ1 + 2|a| and consequently |x+a||x-a|<|x+a|(2|a|+δ1).

Do I have to add an explanation of why I started the new line with |x| - |a| ≤|x-a|< δ1?

Thus for |x^2 - a^2|<ε we have |x-a| < ε/(2|a| + δ1) for |x-a| = min(δ1,ε/(2|a| + δ1).

Anyways some suggestion on how to better improve writing proofs would be appreciated