elementary set theory - How to prove a set is countably infinite

The problem states that A is countably infinite and element b is not in A. It then asks to show that A union {b} is countable infinite.

I'm pretty sure I need to find a bijection between the union and the set of all positive natural numbers, I'm just having trouble figuring out where to go after introducing said function, or how to prove such a function is one to one and onto. Any pointers?

Answer

Let $a_1, a_2, a_3,\ldots$ be a sequence containing all members of the set $A$.

Then $b, a_1, a_2, a_3,\ldots$ is a sequence containing all members of the set $A\cup\{b\}$.

$$
\begin{array}{cccccccccc}
1 & 2 & 3 & 4 & 5 & \cdots \\
\updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \cdots \\
b & a_1 & a_2 & a_3 & a_4 & \cdots
\end{array}
$$