What topology on $\Bbb N$ gives the usual notion of sequential convergence? I know when we talk about limit of function ,there should be with respect to topology on both domain and codomain. And sequence from topological space are basically function from set of natural number to that topological space.My question is.        (1) which topology we assume on set of natural when we are Finding limit of sequence?
 A: By default we don't place any topology on $\mathbb{N}$ when we talk about limits of sequences. If you really want to talk about limits of sequences in this language you can do the following. Take the extended natural numbers $\mathbb{N}_{\infty} = \mathbb{N} \cup \{ \infty \}$ together with the topology generated by the discrete topology on $\mathbb{N}$ and the open sets $\{ n \ge N \} \cup \{ \infty \}$ for all $N$ (this is the one-point compactification of $\mathbb{N}$). Equivalently, this is the subspace topology given by the embedding
$$\mathbb{N}_{\infty} \ni n \mapsto \frac{1}{n} \in [0, 1]$$
where $\frac{1}{\infty} = 0$.
With respect to this topology, finding the limit as $n$ goes to $\infty$ of a sequence of elements of a Hausdorff topological space $X$ is equivalent to finding a continuous extension of this sequence from a function $\mathbb{N} \to X$ to a function $\mathbb{N}_{\infty} \to X$. If such a continuous extension exists, the two definitions of a limit (the usual one for sequences vs. the one for functions between topological spaces) agree, and both agree with evaluation at $\infty$.
