Can you explain neighbourhoods, with respect to the defenition of convergence.

I think of a neighbourhood as a "thick" set containing the point, that is one that extends by at least some fixed amount in every direction around the point. So a one-tenth neighbourhood of 3 on the real line is a set containing 3 and extending at least one-tenth of a unit to either side of 3.

We say that a sequence x_1, x_2, x_3, ... converges to a point p if for each neighbourhood of p there is some number n so that the n_th entry in the sequence, x_n, is in this neighbourhood as well as the k_th entry for every k bigger than n.

I hope this helps

Harley