Exercise 6.5 on p.133 in "Introduction to Graph Theory 5th Edition" by Robin J. Wilson. The definition of complete matching on p.129:

More generally, a complete matching from $V_1$ to $V_2$ in a bipartite graph $G(V_1, V_2)$ is a one-one correspondence between the vertices in $V_1$ and some of the vertices in $V_2$, such that corresponding vertices are joined.

Exercise 6.5 on p.133:

Prove that, if $G = G(V_1, V_2)$ is a bipartite graph in which the degree of each vertex in $V_1$ is not less than the degree of each vertex in $V_2$, then $G$ has a complete matching.  

I cannot solve this problem.
My attempt is here:  
Let $V_1 = \{v_{11}, v_{12}, \cdots v_{1m}\}$.
Let $\deg(v_{11}) \leq \deg(v_{12}) \leq \cdots \leq \deg(v_{1m})$.
Let $V_2 = \{v_{21}, v_{22}, \cdots v_{2n}\}$.  
I think if $\deg(v_{1i}) = 0$ for some $i$, then $G$ has no complete matching.
So, we assume that $1 \leq \deg(v_{11})$.  
We prove this exercise by induction on $m$.  
If $m = 1$, then obviously $G$ has a complete matching.  
$\cdots$ 
 A: Thank you very much for your huge hints, bof.  
We use the Hall's 'marriage' theorem:  

Let $G=G(V1,V2)$ be a bipartite graph, and for each subset $A$ of $V_1$, let $\phi(A)$ be the set of vertices of $V_2$ that are adjacent to at least one vertex of $A$. Then a complete matching from $V_1$ to $V_2$ exists if and only if $|A| \leq |\phi(A)|$, for each subset $A$ of $V_1$.  

Let $V_1 = \{v_{11}, v_{12}, \cdots v_{1m}\}$.
Let $1 \leq \deg(v_{11}) \leq \deg(v_{12}) \leq \cdots \leq \deg(v_{1m})$.
Let $V_2 = \{v_{21}, v_{22}, \cdots v_{2n}\}$.
Let $A = \{w_{11}, \cdots, w_{1k}\}$ be any non-empty subset of $V_1$.
Let $\phi(A) = \{w_{21}, \cdots, w_{2l}\}$.  
Since any edge which is incident on a vertex in $A$ is also incident on a vertex in $\phi(A)$, so $$\deg(w_{11})+\cdots+\deg(w_{1k})\leq\deg(w_{21})+\cdots+\deg(w_{2l}).$$ 
And,  
$$k \deg(v_{11}) \leq \deg(w_{11})+\cdots+\deg(w_{1k}).$$ 
And, 
$$\deg(w_{21})+\cdots+\deg(w_{2l}) \leq l\deg(v_{11}).$$ 
So, $$k \deg(v_{11}) \leq \deg(w_{11})+\cdots+\deg(w_{1k})\leq\deg(w_{21})+\cdots+\deg(w_{2l})\leq l\deg(v_{11}).$$ 
So, $$k \leq l.$$ 
So, $$|A| \leq |\phi(A)|.$$
