Prove that $\beta (G)\le 2\alpha^{'}(G)$ for every graph $G$. For each $k \in \mathbb{N}$ find a graph $G$ with $\alpha^{'}(G) = k$ and $\beta(G) = 2k$ and explain why your graph works.
$\beta(G)$ is the minimum size of a vertex cover
$\alpha^{'}(G)$ is the number of edges in a maximum matching of G
I believe I must prove the first part by induction i.e. doing the case $k=1$ first but i'm not sure how I would actually go about doing this.
No, you don't need to use induction. You should show how you can take a maximal matching of size $\alpha'(G)$ (i.e. comprising $\alpha'(G)$ edges on $2\alpha'(G)$ vertices) and use it to produce a vertex cover of size $2\alpha'(G)$.