Step 1: Assumption: Assume $\sqrt{5}$ is a rational number. Therefore, it can be written as a fraction $\frac{p}{q}$, $q \neq 0$ in its simplest form. This means $p$ and $q$ are integers that share no common factors other than 1, that is, they are co-prime.

Step 2: Square both sides of the equation:

$$5 = \frac{p^2}{q^2}$$

Step 3: Multiply both sides by $q^2$:

$$5q^2 = p^2$$

Step 4: Deduction for $p$: Because $p^2$ is equal to 5 times some integer, $p^2$ must be divisible by 5. If the square of a number is divisible by 5, the number itself must be divisible by 5. Therefore, $p$ is divisible by 5. Let us say $p = 5k$ (where $k$ is an integer).

Step 5: Substitute $p = 5k$ back into our equation from Step 3:

$$5q^2 = (5k)^2$$

which simplifies to:

$$5q^2 = 25k^2$$

Dividing by 5 gives:

$$q^2 = 5k^2$$

Step 6: Because $q^2$ is equal to 5 times an integer, $q^2$ must be divisible by 5, meaning $q$ is also divisible by 5. Since both $p$ and $q$ are divisible by 5, they share a common factor. This contradicts our assumption that $\frac{p}{q}$ is in its simplest form. Because our logical steps are flawless, our initial assumption must be wrong.

∴ $\sqrt{5}$ cannot be expressed as a fraction and is an irrational number.