Proving there is no polynomial algorithm for independent set
Suppose that there exists an integer $K$ and a polynomial time algorithm $A$ which, when run on a graph $G$, outputs a value $A(G)$ which satisfies
$$
 |A(G) - \alpha(G)| \leq K,
$$
where $\alpha(G)$ is the maximum size of an independent set in $G$.

We will show that $A$ can be used to determine $\alpha(G)$ in polynomial time, which contradicts the assumption $\mathsf{P} \neq \mathsf{NP}$.

Given a graph $G$, let $H$ consist of $2K+1$ disjoint copies of $G$; note that $\alpha(H) = (2K+1) \alpha(G)$. Run $A$ on $H$ to get a value $A(H)$ which satisfies
$$
 |A(H) - \alpha(H)| \leq K \Longrightarrow |A(H) - (2K+1) \alpha(G)| \leq K \Longrightarrow \\
 (2K+1)\alpha(G) \in \{A(H)-K,\ldots,A(H)+K\}.
$$
The set on the right contains $2K+1$ values, exactly one of which is a multiple of $2K+1$, hence we can determine $\alpha(G)$ given $A(H)$.