calculus – Theorem for convergence of a series

I was studying Infinite Series. The book I use is Thomas and Finney – Calculus and Analytical Geometry 9th Edition. On pg 635, the following theorem 6 was provided :

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Theorem 6 says that when the sum , say

S = a1 + a2 + a3 + a4 + ….. + a(n)

is continued for infinite terms and converges resulting into a unique and finite value, then the limit of n tending to infinity of a(n) must be zero. This means that as we go further and further with our sum S, we reach to a term that would become, well, very close to zero.

I have tried thinking about it, but I could not really justify this theorem in any manner. The proof of the theorem is not given in the book. Could anyone please help ?