(note the inequality in red; I guess this is what you meant?)

Let so that .

Suppose by contradiction that is not 0 everywhere. Then there is a least index such that , and we can factor the sum to write , so that you can see we have

This limit is also the limit along any sequence that converges to 0. However, you know there is such a sequence where is zero. We deduce from the above limit that , a contradiction.