Q:

If a series converges, then the sequence of terms converges to $0$. Why?

I am an AP Calc BC student who has just recently been introduced to sequences and series.
The proof from the checked answer to this question makes sense:
If a series converges, then the sequence of terms converges to $0$.
However, it's still not clicking intuitively, and I'd really like an intuitive understanding of why this is so.
Why does this make intuitive sense?

A:

If you're looking for an intuitive, non-technical reason, here you go...
Think of it like walking along the number line. Each term in the series tells you how big of a step to take in the positive / negative direction. If a series converges, then your steps will pull you into one specific value, to which you get "infinitely close." The only way you can get close to this value after taking an arbitrarily large number of steps is if your individual steps get really small. 

