If a sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence {\displaystyle (a_{n})} is normally denoted {\displaystyle \lim _{n\to \infty }a_{n}} . If {\displaystyle (a_{n})} is a divergent sequence, then the expression {\displaystyle \lim _{n\to \infty }a_{n}} is meaningless.
is defined as the set of all sequences {\displaystyle (x_{i})_{i\in \mathbb {N} }} such that for each i, {\displaystyle x_{i}} is an element of {\displaystyle X_{i}} . The canonical projections are the maps pi : X → Xi defined by the equation {\displaystyle p_{i}((x_{j})_{j\in \mathbb {N} })=x_{i}} . Then the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections pi are continuous. The product topology is sometimes called the Tychonoff topology.