Find the least upper bound (LUB) and greatest lower bound (GLB) of $\{x\sin(1/x):x>0\}$
My Attempt: Since limit of given function is $0$ as $\lim_{x\to f(x)}g(x) = 0$ when $f(x)\to0$ and $g(x)$ is bounded. So $\operatorname{LUB}(f) = \operatorname{GLB}(f) = 0$ But it don't resemble with actual answer. Please help me.