 Solution

The coefficient sum is finite, so the Weierstrass M-test makes the cosine series uniformly convergent and $f$ continuous. Put $Q=5^{m+1}$ and $A_m={\sum }_{k=m+1}^{\infty }{a}_k$. The indicated partial sum belongs to ${\mathcal{T}}_n$ because its largest frequency is $5^m\le n$. Its tail satisfiesand equality holds at $x=0$. Thus its supremum norm error is exactly $A_m$.

At the $2Q$ points $x_j=j\pi /Q$, $0\le j<2Q$, every omitted frequency is an odd integer multiple of $Q$. Indeed for $k\ge m+1$,Uniform convergence permits summing these values, givingThese are cyclically alternating extrema. Since $n<Q$, there are $2Q\ge 2n+2$ of them, so the trigonometric Chebyshev alternation theorem proves that the partial sum is the unique best approximant. Consequently the positive lacunary trigonometric series hasThe positivity of every tail coefficient makes the same tail attain its full coefficient sum with alternating signs; a triangle bound alone would not establish optimality.