Linear Function is Continuous/Proof 2

Proof
Let $c \in \R$.

Let $\sequence {x_n}$ be a real sequence converging to $c$.

Then:

We therefore have:


 * for all real sequences $\sequence {x_n}$ converging to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.

So by Sequential Continuity is Equivalent to Continuity in the Reals $f$ is continuous at $c$.

As $c \in \R$ was arbitrary, $f$ is continuous on $\R$.