I want to prove that for a metric space $(\frak{M},\rho)$ if $M\subset \frak M$ satisfies $$M\subset \mathop{\bigcup}_{k=1}^n B[x_k,r_k]\;\;(n\in\mathbb N),$$ then $M$ is a bounded set.
Here's my attempt at proving this:
Take balls $B_1=B[x_1,r_1], B_2=B[x_2,r_2]$. Let $x\in B_1, x'\in B_2$. Then, from triangle inequality, we have $$\rho(x,x')\le\rho(x,x_1)+\rho(x_1,x')\le\\ \le\rho(x,x_1)+\rho(x_1,x_2)+\rho(x_2,x')\le\\ \le r_1+\rho(x_1,x_2)+r_2.$$
Then for a ball $B'=B[x_1,r_1+r_2+\rho(x_1,x_2)]$ we have $$B'\supset B_1\cup B_2.$$
Now, iteratively considering balls $B'$, $B_m$ for $2<m\le n$, we'll get bigger ball $B^{(m)}$, including more balls $B_i, i\le m$, until we stop at $m=n$, where our biggest ball is still finite. Thus, since a union of two balls is bounded, then union of any finite number of balls is bounded, and thus $M$ is also bounded, QED.
The question
Now, what bothers me is that usually "and so on" arguments similar to the paragraph above are done using proof by induction. On the other hand, such boundedness problem is really just about finite number of identical operations. So, the question is: is my argument rigorous enough, or do I still have to do proof by induction? If induction is mandatory here, then are there any examples where my argument would fail?
Your argument is fine, and it works by induction. A union of $n$ balls is the union of $n-1$ balls and $1$ ball, and the union of $n-1$ balls is contained in a ball by induction. So the argument reduces to the case of $2$ balls. So one way to interpret your proof is that you are proving your statement for any finite $n$ by induction. It does not say that the argument holds for an infinite union of balls.