I'm trying to prove that if there is an injection $f:\lambda\to\kappa$ (for $\lambda$,$\kappa$ cardinal numbers) then $\lambda\leq\kappa$. This is not true if they are just ordinal numbers, for example it is easy to build an injection from $\omega+1$ to $\omega$, however $\omega<\omega+1$.
I think the proof should be quite straightforward but I'm not getting it.
I want to arrive to a contradiction by assuming $\kappa<\lambda$, so $f|_\kappa:\kappa\to\kappa$ is an injection and $f(\kappa)$ is propper subset of $\kappa$ (not necessarly an ordinal number) but I don't realize how this can be problematic or how to move from here.
I thought maybe I should well order $f(\kappa)$ (and for this I think I need AC) and do something with its order type, but again I'm not sure how to proceed.