Thread: limit superior inequality

Let an, bn be two bounded sequences. Show that

limn-->∞ sup (an+bn) ≤ limn-->∞ sup an + limn-->∞ sup bn

That's because . Indeed, on the right you can, roughly speaking, choose the largest elements of the sequences and independently and then add them, whereas on the left you have to add elements with the same index and then choose the largest.