Why is this a proof of the Boundedness Theorem?

Solution 1:

You find a contradiction because you are assuming that $|f(x_n)|$ diverges to infinity; but then, you extract a convergent subsequence of $(x_n)$, namely $(x_{n_k})$, converging to some $c$ in $[a,b]$. Because $f$ is continuous, $|f(x_{n_k})|$ converges to $|f(c)|$, which is a totally well defined number in $\mathbb{R}$ (because $f(c)\in\mathbb{R}$), i.e. does not explode to infinity. To conclude, remember that if the limit of a sequence exists, then the limit of a subsequence is the same as the limit of the sequence. Therefore, in the end you say that $|f(x_n)|$ converges to $|f(c)|$, which is the contradiction you were looking for