Continuous Functions on ${{\textbf{R}}}$

Abstract

In previous chapters we have been focusing primarily on sequences. A sequence $(a_n)_{n=0}^\infty$ can be viewed as a function from ${{\textbf{N}}}$ to ${{\textbf{R}}}$, i.e., an object which assigns a real number $a_n$ to each natural number n. We then did various things with these functions from ${{\textbf{N}}}$ to ${{\textbf{R}}}$, such as take their limit at infinity (if the function was convergent), or form suprema, infima, etc., or computed the sum of all the elements in the sequence (again, assuming the series was convergent).