Successor of Ordinal Smaller than Limit Ordinal is also Smaller/Proof 1

Proof
Let $\lambda$ be a limit ordinal such that $\alpha < \lambda$.

From Successor of Element of Ordinal is Subset

Then as $\alpha^+$ is the successor set of $\alpha$ it follows that:
 * $\alpha^+ \le \lambda$

But as $\lambda$ is not a successor ordinal:
 * $\alpha^+ \ne \lambda$

Hence:
 * $\alpha^+ < \lambda$