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Bilateral graph bound proof

A friend presented me with a hypothesis that seems true, but none of us can come up with evidence. Here's the problem:

Given a connected bipartite graph with disjoint nonempty vertex sets U and V, such that | U | & lt | | V |, all vertices are either in U or in V, and there are no edges connecting two vertices within the same set, then there is at least one edge that connects the vertices a∈U and b∈V such that the degree (a)> degree (b)

It is trivial to prove that there exists at least one vertex in U with a degree higher than one in V, but to neglect the pair exists with an edge attached to it.

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e = (a, b) a∈U b∈V, w (e) = 1/deg (b) -1/deg (a). x 1/deg (x) , x, 1, deg (x) . , w (e) e | V | - | U |. | V | - | U | > 0, w (e) > 0 e = (a, b), , deg (a) > deg (b).

+2

, .. , deg (a) ≤ deg (b) ∀ (a, b) ∈ E, E - ( , U V).

F⊆E V (F) V, F, :

V (F) = {b | (a, b) ∈ F}

F :

F = empty set
For a ∈ U:
    add any edge (a,b)∈E to F
Keep adding arbitrary edges (a,b)∈E to F until |V(F)| = |U|

V (F) U,

Σ a∈U deg (a) ≤ Σ b∈V (F) deg (b)

, | U | = | V (F) | | U | < | V | , "" node v∈V\V (F), , deg (v) > 0,

Σ a∈U deg (a) Σ b∈V deg (b)

; .

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