### Bound on norm of unit vector (PSET 8, ex 2 part 1)

Hello, I have a question regarding the following bound on the norm:

if $$u = \sum_i c_i u_i$$, then $$||u||^2 = \langle u, u \rangle = \langle \sum_i c_i u_i, \sum_j c_j u_j \rangle = \sum_{i, j} c_i c_j \langle u_i, u_j \rangle = \sum_i c_i^2$$ if we assume that the $$\{u_i\}$$ forms an orthonormal basis. Hence I don't understand why there is an inequality in the solution. Where is the mistake in my reasoning ? (Note: $$c_i$$ is the projection of $$u$$ on $$u_i$$ in the solution)