How to show that this set is finite?

Question

Let $m \in \mathbb{N}$
For $\alpha = (\alpha_{1},...,\alpha_{m}) \in \mathbb{N}_{0}^{m}$, let $|\alpha|:= \alpha_{1}+...+\alpha_{m}$
Is the set $\{\alpha \in \mathbb{N}_{0}^{m}: |\alpha|\leq k\}$ finite for any $k \in \mathbb{N}$? I'm almost sure that it is but I have no idea how to prove it or even give a convincing explanation as to why that's the case. I thought maybe use proof by induction but the argument seems to be very complicated. Is there a contradiction you can reach by assuming the set to be infinite?
Apologies for tags, I don't know which section to post this question in.

Answer 1

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.

Answer 2

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 $\alpha_i\ge k+1$.

Answer 3

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 $a_i \leq k$.