Could you guys please help me understand (intuitively) the definition of limit of a sequence? This is the definition I have:
I'm trying to visualize what's going on, I imagine two parallel lines, lower one representing the domain $\lambda$ (which is a limit ordinal) whose elements are $\gamma$, and the upper one representing the codomain, the line containing all the $f(\gamma)=\beta_{\gamma}$.
Now the definition says:
"We say that the ordinal $\alpha$ is the limit of the sequence.." where should I put this $\alpha$? Is it part of the domain or of the codomain? I think it's part of the codomain so I put a dot on the codomain line to represent this $\alpha$.
"..if for each $\delta<\alpha$.."
since $\alpha$ was on the codomain line I put this $\delta$ to the left of this $\alpha$ on this same line.
"..there is some $\gamma_{\delta}<\lambda$.."
since we're talking about ordinal numbers $<$ means $\in$, therefore I put $\gamma_{\delta}$ on the domain line,
"..such that for all $\gamma$ with $\gamma_{\delta}<\gamma<\lambda$.."
this means, such that for every number between this last dot we drew and $\lambda$ (which is the last number of the domain)
"..$\delta<\beta_{\gamma}\leq\alpha$"
this means there is some number of the sequence between $\alpha$ and $\delta$
Is representing this definition in this way correct?
Almost correct. It gets a bit dodgy at the end. First, $\lambda$ is not the last ordinal in the domain, it is the domain. Since these are ordinals that means it is the first number not in the domain. More importantly, the last part should be "every [corresponding] number of the sequence is between $\alpha$ and $\delta$". If we imagine $\gamma$ moving along the line from 0 towards $\lambda$ then the corresponding $\beta_\gamma$ can jump around anywhere on the codomain line to start with, but there comes a point after which it stays above $\delta$, and that applies however close we choose $\delta$ to $\alpha$. (It is very analogous to the $\epsilon-\delta$ definition in analysis.)