Let {$a_n$} and {$b_n$} be two sequences of real numbers such that {$b_n$} converges to L. It is known that the supremum S of the set {$a_n$: n $\in$ N} exists and is not equal to any of the $a_j$'s. Define the sequence {$c_n$} by $c_n$ = $a_n$ + $b_n$ for each n $\in$ N. Prove that $\limsup_{x\to∞}$ $c_n$ = S + L .
I encountered this problem from an analysis course. I tried to use the fact that $b_n$, a converging sequence, is bounded and since $b_n$ is a real sequence, it has supremum S. Next,
Since $c_n$ = $a_n$ + $b_n$, $b_n < L$ and $a_n < S$
${c_n} < L + S$.
However, I cannot prove that L + S is the supremum of $c_n$.