Solution
= Solution
\b[False.] If <uniform convergence> held, some <polynomial> $P_n$ would satisfy $\sup_{t\in\mathbb R}|P_n(t)-\sin t|<1$. Such a <polynomial> would be bounded on $\mathbb R$, hence constant: a nonconstant <polynomial> grows without bound in absolute value along a real half-line. For any constant $c$, however,
$$
\sup_{t\in\mathbb R}|c-\sin t|\geq\max\{|c-1|,|c+1|\}\geq1,
$$
a contradiction.