Can any infinite ordinal be expressed as the sum of a limit ordinal and a finite ordinal?

I've been browsing through Jech's and Levy's texts on set theory, and the ideas of ordinals come up fairly quickly. The idea of a limit ordinal is introduced, which is an ordinal with no maximum element. My question is, can any infinite ordinal be written as the sum of a limit ordinal and a finite ordinal, possibly unique?

My thinking was, if $\alpha$ is an infinite ordinal with no maximum, it is a limit ordinal, so $\alpha=\alpha+0$. Otherwise, suppose $\alpha$ has some order type $\{a_0,a_1,\ldots, b\}$, so $\alpha=\omega+1$. Similarly, if $\alpha$ has order type $\{a_0,a_1,\ldots,b,c\}$, we could write it as $\omega+2$.

(Thanks to Arturo Magidin, for pointing out that the following example I gave is not an ordinal.) But what about an order type like $\{a_0,a_1,a_2,\ldots, b_2,b_1,b_0\}$, this has order type $\omega+\omega^*$, would it still be possible to write is a sum of a limiting ordinal and a finite ordinal? Thanks.

This is true by a simple (complete) induction:

Case I: $\alpha$ is limit, vacuously true (as you observed)

Case II: $\alpha=\beta+1$, then $\beta=\beta'+n$ and thus $\alpha=\beta'+(n+1)$.

This of course can be expanded to inversed ordinals as well, resulting that every order of the form $\alpha^*+\beta$ can be written as a sum of two limit ordinals (one which is the inverse of an ordinal, to be accurate) and two finite ordinals (again, one is inverse of a finite number).