Show that $5 + 2\sqrt{7}$ is an irrational number, where $\sqrt{7}$ is given to be an irrational number.

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Step 1: Let us assume that $5 + 2\sqrt{7}$ is not an irrational number. Then $5 + 2\sqrt{7}$ is a rational number, say $p$. So $5 + 2\sqrt{7} = p$.
Step 2: $\Rightarrow \sqrt{7} = \frac{p - 5}{2}$
Step 3: Since $p$ is rational, $\frac{p - 5}{2}$ is also rational. This is a contradiction as RHS is rational but LHS ($\sqrt{7}$) is irrational.
Step 4: Hence $5 + 2\sqrt{7}$ cannot be rational, so it is irrational.

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