Topology/Sequences

A sequence in a space is simply a function from the natural numbers into the space, $f:N->X$. The general idea is that each point in the space is associated with a number; the first element of the sequence is $f(1)$, the next is $f(2)$, etc. For example, consider the sequence in $R$ given by $f(n)=1/n$. This is simply the points $1,1/2,1/3,1/4,...$ Also, consider the constant sequence $f(n)=1$. You can think of this as the number 1, repeated over and over.

Often, sequences are denoted with subscripts. For example, the sequence defined in $N$ by $f(n)=2n$ is the sequence $2,4,6,8,\dots$, and can also be denoted $a_n=2*n$. Sequences themselves where f(i)=a_i can be denoted {a_i}.

Convergence

A sequence {x_i} is said to converge to x if for any neighborhood of x, there exists an N such that for any n>N, x_n is within that neighborhood.

Exercises

1. give a rigorous description of the following sequences of numbers:
   • $1,2,3,4,5\dots$
   • 2,-4,6,-8,10,...
   • 1,10,100,1000,10000,...