= Solution
Use one orthonormalized localized <atomic orbital> $|j\rangle$ at each position $ja$. In the nearest-neighbour <tight-binding model>, translation invariance gives the on-site energy and real <hopping integral>
$$
E_0=\langle j|H|j\rangle,\qquad
t=\langle j|H|j+1\rangle=\langle j+1|H|j\rangle.
$$
The <Bloch state> is
$$
|k\rangle=\frac1{\sqrt N}\sum_j e^{ikja}|j\rangle.
$$
Since $H|j\rangle=E_0|j\rangle+t|j-1\rangle+t|j+1\rangle$, shifting the two summation indices gives
$$
H|k\rangle=[E_0+te^{ika}+te^{-ika}]|k\rangle,
\qquad\boxed{E(k)=E_0+2t\cos(ka).}
$$
The <hopping integral> is an energy, and its sign sets whether $k=0$ is a minimum or maximum. $E_0$ is the diagonal matrix element in the crystal Hamiltonian, which can differ from an isolated-atom energy.
The <orthogonal tight-binding approximation> matters. If raw neighbouring orbitals retain a nonzero <orbital overlap> $s$ in their normalization, projection instead gives a <generalized eigenvalue problem> and the <nonorthogonal nearest-neighbour tight-binding band> $E(k)=[E_0+2h\cos(ka)]/[1+2s\cos(ka)]$. The stated cosine band follows after orthonormalization and nearest-neighbour truncation, or after neglecting that overlap metric while retaining the hopping it motivates.
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