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I do not understand the emphasized inequality: http://s11.postimage.org/gnnf1yirn/Capture.png

How is it that if the size of X is greater than n, then the number of its neighbours is greater or equal to the neighbours of S?

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up vote 2 down vote accepted

Since $S\subset X$, $N(S)\subset N(X)$ and hence $\vert N(S)\vert\leq \vert N(X)\vert$.

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