Upper Bound of Ordinal Sum

Theorem
Let $x$ and $y$ be ordinals.

Suppose $x > 1$.

Let $\langle a_n \rangle$ be a finite sequence of ordinals such that:


 * $a_n < x$ for all $n$

Let $\langle b_n \rangle$ be a strictly monotonic finite sequence of ordinals such that:


 * $b_n < y$ for all $n$

Then:


 * $\displaystyle \sum_{i = 1}^n a_i x^{b_i} < x^y$

Proof
The proof shall proceed by finite induction on $n$:

For all $n \in \N_{\le 0}$, let $P \left({n}\right)$ be the proposition:
 * $\sum_{i = 1}^n a_i x^{b_i} < x^y$

Basis for the Induction

 * $P\left({ 0 }\right)$ says that:


 * $\sum_{i = 1}^0 a_i x^{b_i} < x^y$

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\sum_{i = 1}^k a_i x^{b_i} < x^y$

Then we need to show:
 * $\sum_{i = 1}^{k + 1} a_i x^{b_i} < x^y$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\sum_{i = 1}^n a_i x^{b_i} < x^y$