For my intro to DSP class, the TA gave us this discrete system and asked to prove whether or not it's BIBO-stable:
$T \left\{ x[n] \right\} = \sum \limits_{k=\text{min}(n, n_0)}^{\text{max}(n, n_0)} {x[k]}$
for some integer constant $n_0$.
Intuitively, it's unstable, and it can be easily proven by a counterexample if $x[n]$ is the unit step and $n_0 = 0$.
However, the TA tried to use a general proof by invoking the triangle inequality. Assuming $x[n]$ is bounded (or $|x[n]| \leq M < \infty$), he said:
$\left| T \left\{ x[n] \right\} \right| = \left| \sum \limits_{k=\text{min}(n, n_0)}^{\text{max}(n, n_0)} {x[k]} \right| \leq \sum \limits_{k=\text{min}(n, n_0)}^{\text{max}(n, n_0)} {|x[k]|} \leq (|n-n_0|+1)M$
Obviously the right-hand side is unbounded, as it goes to infinity with increasing $n$, but to me it doesn't seem to imply that the system on the left-hand side is unbounded (because of the inequality).
My question is, can his attempt be augmented to show that the left-hand side is also unbounded? Or is a counter-example the only way to prove it?