It is easy to prove the limit exists, all we have to show is there exists a relationship between $\\delta$ and $\\epsilon$. But how are we supposed to prove limit doesn't exists? The problem is when ...
I'm trying to teach myself how to do $\\epsilon$-$\\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different. Exe...
I would like someone to state the steps and associated reasoning involved in proving that a sequence converges, using the epsilon definition of convergence. Please specify the reasoning behind each step of the methodology, to assist in justifying your calculations. I would like the 'why' and 'how' behind each step of such a proof.
There are three ways to interpret this question. You could be asking "what is the difference between the two directions of implication?". You could be asking "what, intuitively, is the property of continuous functions captured by the real definition that is not captured by the backwards one?" You could be asking "what are the consequences of the backwards definition?" The answer to question 1 ...
As in most $\epsilon-\delta$ proofs, we start at the inequality we want to be true, then work backwards to find the necessary restrictions on $\delta$. Then we present the forwards implications using the found $\delta$. First, let us rewrite the inequality in polar coordinates.
I know, this question is answered. But would it have been okay to use a series argument and then say: Well, there is a theorem which tells us $\varepsilon-\delta$-continuity is the same as sequence-continuity. So there has to be a $\delta \gt 0$ fulfilling your requirements.
