for some integer constant .
Intuitively, it's unstable, and it can be easily proven by a counterexample if is the unit step and .However, the TA tried to use a general proof by invoking the triangle inequality. Assuming is bounded (or ), he said:
Obviously the right-hand side is unbounded, as it goes to infinity with increasing , 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?