Oh, so the first line of a complete proof should read:
\lim_{x\to a} f(x) = L_1 \iff \forall ε > 0 \ \exists δ_1 > 0 \ : \ 0 < |x - a| < δ_1 \implies |f(x) - L_1| < ε
And the rest would have to be corrected analogously. Is it correct now, or am I still missing something?
Yes, I understand that the proof is not complete; I omitted the quantifiers for the sake of brevity. My question was intended to be more in line of 'Is my reasoning correct?' than 'Is the proof a complete, rigorous proof?', i.e. is all I have to do from that point on is add the correct...
This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate.
Homework Statement
Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h)
Homework EquationsThe Attempt at a Solution
By the...