How to show that this set is finite?

Solution 1:

We have $$ \{\alpha\in\Bbb N_0^m : |\alpha| \leq k\} \subseteq \{\alpha\in\Bbb N_0^m : (\forall i) \alpha_i \leq k\} = \{1,\ldots, k\}^m $$ and the letter set is indeed finite, as there are only finitely many natural numbers $a_i \leq k$.

Solution 2:

If $|\alpha|\leqslant k$, then, for each $i\in\{1,2,\ldots,m\}$, $\alpha_i\leqslant k$, and therefore your set is a subset of $\{0,1,\ldots,k\}^m$, which is finite.

Solution 3:

Hint

Prove that if the set is infinite, there must be an $\alpha\in \Bbb N_0^m$ for which, there exists $\alpha_i$ such that $\alpha_i\ge k+1$.