Question

Is there any proof or theorem that says "The eigenvector space of a matrix A is <= n " for a n x n matrix A ??

Answer 1

indirectly spokens yes. A nxn matrix can only have n or less non-similar eigenvalues. Each eigenvalue has one "partner" eigenvector. So there can only be n or less independent eigenvectors. (Think about diagonalisation)
Note: If you hear somebody say "THE eingenvector to this eigenvalue", they mean one REPRESENTATIVE of all linear combinations of this eigenvector.