Continuity Theorem Addition Proof

Prove that if $f$ is cont at $a$, and $g$ is cont at $a$, then $f+g$ is cont at $a$

Proof

1. Assume $f$ is cont at $a$
2. Assume $g$ is cont at $a$
3. Then ${\lim }_{x\to a}f(x)=f(a)$
4. Then ${\lim }_{x\to a}g(x)=g(a)$
5. Then ${\lim }_{x\to a}f(x)+g(x)=f(a)+g(x)$ limit sum rule
6. by definition $f+g$ is continuous at $a$