Infinite sequences of natural numbers with finite support are equivalent to $\omega^{n_1} c_1 + \omega^{n_2} c_2 + \ldots + \omega^{n_k} c_k$

Question

Why exactly do infinite sequences of natural numbers with finite support (having a finite number of non-zero values) correspond to ordinals of the form $\omega^{n_1} c_1 + \omega^{n_2} c_2 + \ldots + \omega^{n_k} c_k$? I found it mentioned on the Wiki page on ordinal exponentiation. Is it possible to explain from the first principles that ordinals of the above form are actually sequences with finite support?

Answer 1

I suppose that the sequence $(a_0,a_1,a_2,\ldots,a_n,0,0,\ldots)$ corresponds to the ordinal $$\omega^n a_n+\omega^{n-1} a_{n-1}+\cdots+\omega a_1+a_0.$$