= Unitary sphere quotient and stability range
{title2=$U(n)/U(n-1)\cong S^{2n-1}$}
The last-column map has fibre the subgroup fixing the last standard unit vector. Unitary completion gives surjectivity, and <compactness> gives the quotient <homeomorphism>. Projecting the other standard vectors onto the <orthogonal complement> of a nearby unit vector and applying Gram-Schmidt gives a local section, so the projection is a fiber bundle. The bundle $U(n)\to U(n+1)\to S^{2n+1}$ then gives $\pi_iU(n)\cong\pi_iU(n+1)$ for $i\le2n-1$ because both adjacent <sphere> <homotopy groups> vanish.
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