So I've been given the following
R = Map(ℝ, ℝ) with addition and multiplication defined by $(f +g)(x) = f(x)+g(x)$ and $(f ·g)(x) = f(x)g(x)$. Let S be the ring of sequences $(a_{n})_{n}≥0$ with entries $a_{n} ∈ ℝ$, and define
φ:R→S
$f → (f(n))_{n}$
I need to use the first isomorphism theorem to find the isomorphism with the domain R/I (where I is the kernel of φ) and codomain S. I have already found I, and have been given the hint that I need to show that the homomorphism given is surjective but I'm not sure how to do that.
I know that for the definition of surjectivity is that for every element of S, there is an element of R that maps to it, but I have no idea how to show that there is a function $f$ that maps to a sequence of the form $(f(n))_{n}$
When in doubt, expand definitions! Write out explicitly what surjectivity says in this case:
For every sequence $(a_n)_{n \in N} $, there is a function $f : R \to R$ such that $(f(n))_{n \in N} = (a_n)_{n \in N}$.
And that equality can then also be written as a more explicit condition:
For every sequence $(a_n)_{n \in N} $, there is a map $f : R \to R$ such that for all $n \in N$, $f(n) = a_n$.
Here it depends what you meant by $Map(R,R)$ at the beginning: arbitrary functions, or continuous, or smooth…? This fact holds for any of those choices, but is easier to show the fewer requirements there are on $f$.
(Please excuse bad LaTeX; I’m typing on mobile.)