Removing "i" from the triangle inequality?

Question

In my complex analysis class, I have to prove that a limit exists for a function and I think I can use the Triangle Inequality in my proof, but I don't know if it's possible. My question: knowing that $|a| + |b| \le |a + b|$, am I allowed to then say $|a| + |b| \le |a + ib|$? I saw someone do it in their own proof, but I'm a bit skeptical mostly because I don't quite understand. Thanks!