First of all let's we remember the following result.
Theorem
Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ in $\alpha$ implies that for $\alpha$.
Let be now $f$ a function from an order $(X,\mathcal U)$ into an order $(Y,\mathcal V)$: we will say that $f$ is monotone if for any $a$ and $b$ in $X$ the inequality $$ a\preccurlyeq_\mathcal U b $$ implies the inequality $$ f(a)\preccurlyeq_\mathcal V f(b) $$ So let's we prove the following result.
Conjecture
A function $f$ from an ordinal $\lambda$ into an order $(X,\mathcal O)$ is monotone if and only if the following holds:
- for any $\delta$ in $\lambda$ the inequality \begin{equation} f(\delta)\preccurlyeq_\mathcal O f(\delta+1) \tag{1} \label{successive imagine} \end{equation} holds;
- if $\alpha$ in $\lambda$ is limit then the inequality \begin{equation} f(\beta)\preccurlyeq_\mathcal O f(\alpha) \tag{2} \label{limit imagine} \end{equation} holds for any $\beta$ in $\alpha$.
Proof Let's we assume that ineq. \eqref{successive imagine}-\eqref{limit imagine} hold and thus let's we prove even the proposition \begin{equation}(\forall\alpha)\Biggl((\alpha\in\lambda)\to\biggl((\forall\beta)\Big((\beta\in\alpha)\to\big(f(\beta)\preccurlyeq_\mathcal O f(\alpha)\big)\Big)\biggl)\Biggl) \tag{3} \label{monotonia} \end{equation} holds. So let be $\alpha$ an element of $\lambda$ and thus let's we assume the proposition \begin{equation} (\forall\gamma)\Big((\gamma\in\beta)\to\big(f(\gamma)\preccurlyeq_\mathcal O f(\beta)\big)\Big) \tag{4} \label{ipotesi induttiva} \end{equation} for all $\beta$ in $\alpha$. Well if there exist an ordinal $\delta$ such that $$ \alpha=\delta+1 $$ then any for any $\beta$ in $\alpha$ the inequality $$ \beta\le\delta $$ holds so that by inductive hypothesis and ineq. \eqref{successive imagine} the inequality $$ f(\beta)\preccurlyeq_\mathcal O f(\delta)\preccurlyeq_\mathcal O f(\delta+1)=f(\alpha) $$ holds; after all if $\alpha$ is limit then by ineq. \eqref{limit imagine} surely \ref{ipotesi induttiva} holds for $\alpha$ too: so we conclude \ref{ipotesi induttiva} holds for $\alpha$ always and thus by transfinite induction it generally holds for all $\alpha$ in $\lambda$ so that \ref{monotonia} holds.
Conversely if $f$ is monotone then ineq. \eqref{successive imagine}-\eqref{limit imagine} trivially holds -by monotonicity definition.
So I ask if the conjecture is actually true and thus if I well proved it: could someone help me, please?