 Solution

Every derivative of order $n+1$ of $\sin (2x)$ has absolute value at most $2^{n+1}$. The equally spaced nodes are symmetric about $1/2$. Pair each node $a\le 1/2$ with $1-a$; for $x\in [0,1]$,If an unpaired middle node remains, its factor satisfies $|x-1/2|\le 1/2$. Consequently the complete nodal product is at most $2^{-(n+1)}$. The error formula actually gives the stronger estimateThe first bound tends to zero, proving uniform convergence on the whole interval. There is no appeal to a general claim that equispaced interpolation always converges; the derivative and symmetric-product bounds are particular to this analytic function and these nodes.