Finding limits superior and inferior of a sequence

Question

Find the limits superior and inferior of the following sequences: note: "For a set, those are the infimum and supremum of the set's limit points, respectively. "

$a_n=\frac{n}{n+1} \sin{\frac{2n\pi}{4}}$

and

$c_n=1+2(-1)^{n+1}+3(-1)^{\binom{n}{2}}$ detailed help appreciated, because i do not know much

Answer 1

For the first sequence $\frac{n}{n+1} \longrightarrow 1$ as $n \rightarrow \infty$ and $sin(2n \pi /4)$ fluctuates between 1 and -1

therefore limsup = 1 and liminf = -1

for the second sequence the supremum is 1 + 2 + 3 = 6 the infimum is 1 - 2 -3 = -4

as the terms subjected to n are either 1 or -1 hence the simple upper and lower bounds

hope this helped