Thread: Definition of a sequence

My book has the attached definition. Could someone explain to me in English what the bottom part is saying please?

Any input would be greatly appreciated!
Thanks in advance!

The intuitive meaning of "a sequence a_n converges to a limit L" is that the farther out in the sequence you look, the closer the values get to L, and more specifically, if we look far enough out in the sequence, the values get as close as we want to L (and stay at least that close forever after).

The integer N in the definition represents how far out we're looking in the sequence--that's why it says "when n>N". What the definition is really saying is: "tell me how close you want the values of the sequence to be to L, and I'll give you a number N, so that every value, after the Nth value, is at least that close to L".

What about that inequality with the epsilon though?

Originally Posted by s3a

What about that inequality with the epsilon though?

The epsilon is how close you need the sequence to be to L. The definition is saying we can pick any positive epsilon arbitrarily small and we can find an N so that when we get to n>N in the sequence, the sequence is closer than epsilon to L.