Let $\{u_k\}_{k=1}^\infty$ be a sequence of ($L^2$ normalized) mutually orthogonal eigenfunctions of the operator $-\Delta$ on the sphere $\mathbb{S}^n$ (here $\Delta$ is the Laplace Beltrami operator). Let $u$ be a smooth (real valued) function on the sphere. It is a well-known result that we can write $u=\sum_{k=1}^\infty c_k u_k$ for some (real) constants $c_k$. My question is: Is the convergence of this sum uniform?

I am trying to prove that the optimal constant in the Poincare inequality is $\lambda_1=n$. That is to say, I am trying to prove the inequlity $\int_{\mathbb{S}^n} |\nabla u|^2 \geq n \int_{\mathbb{S}^n} |u|^2$. Here is what I have done so far:

First, integrate by parts on the LHS so that it suffices to prove $\int_{\mathbb{S}^n} -u\Delta u \geq n \int_{\mathbb{S}^n} |u|^2$. Then use $u=\sum_{k=1}^\infty c_k u_k$ and *assume that the convergence is uniform*. Then we can switch the order of the sum with the derivative and integral (and use the fact that $\{u_k\}$ are orthonomal) so that
\begin{align*}
\int_{\mathbb{S}^n} -\left(\sum_{k=1}^\infty c_k u_k\right)\Delta \left(\sum_{j=1}^\infty c_j u_j\right)&=
\int_{\mathbb{S}^n} -\left(\sum_{k=1}^\infty c_k u_k\right)\left(\sum_{j=1}^\infty c_j \Delta u_j\right)=
\int_{\mathbb{S}^n} -\left(\sum_{k=1}^\infty c_k u_k\right)\left(\sum_{j=1}^\infty \lambda_j c_j u_j\right)
\\&= \sum_{j,k}c_k c_j \lambda_j\int_{\mathbb{S}^n} u_k u_j= \sum_{j,k}c_k c_j \lambda_j \delta_{jk}=\sum_{j}c_j^2 \lambda_j\geq \lambda_1 \sum_j c_j^2.
\end{align*}
By the same logic, the last sum is equal to $\int |u|^2$.

Now obviously, this proof requires some argument showing that the sum commutes with $\Delta$ and the integral but I have not been able to find a reference that the sum converges uniformly. My thought is that this would follow from some basic facts in Harmonic analysis though I am no expert in that field. Would anyone be able to provide a reference for this?