If we are given a series from n to infinity and know it is non-negative and convergent, is it also decreasing?

Question

I was given a $\ \sum_{n}^{\infty} a_{n}$ and know that it is non-negative and convergent. I was curious since we know its convergent and by the necessary/sufficient condition its limit as n goes to infinity is 0, is the sequence {${a_{n}}$} decreasing? More explicitly is it true that ${a_{n}}$ > ${a_{n+1}}$? I can only think of the sequence 1 + 1 + 0.5 + 0.5 + 0.25 + 0.25 ... as a counterexample but I'm not sure if its a good one. Thank you