(a) Let $G=( X \cup Y, E)$ be a connected bipartite graph. Show that every edge of $G$ extends to a matching from $X$ and $Y$ if and only if for all $A \subseteq X$,$A\neq \emptyset ,X$, we have $\left|N_G (A)\right| \gt \left|A\right|$.
(b) Let $K=( X \cup Y, E)$ be a infinite bipartite graph satisfying Hall's condition (that is, $\left|N_G(A)\right| \geq \left|A\right|$ for every $A \subset X$). Show that it does not necessarily contain a matching from $X$ to $Y$.
For part (a), I tried to do the $(\Rightarrow)$ part (not sure it's correct or not) but I couldn't do the $(\Leftarrow)$ part. I know it should be similar to the prove of Hall’s marriage theorem by induction. Below is my answer for $(\Rightarrow)$.
As $G$ extends to a matching from $X$ to $Y$, then all the vertices must have degree $\gt$ 1, therefore it implies that $|N_G (A)| \gt |A|$ for all $A \subset X$.
And for part(b), I have totally no clue how to start it. Could anyone teach me how to do it please?