Suppose $x_n$ is a bounded sequence and $\limsup x_n = \liminf x_n = c$. Prove $x_n \rightarrow c, n \rightarrow \infty$
Not sure if there is a better way to show this, or if my way is even correct, just looking for some guidance or tips.
Since $\limsup x_n = c$, then given $\epsilon > 0$, there exists a $N_1$ such that for all $n \geq N_1$ we have:
$x_n < c + \epsilon$.
Likewise, since $\liminf x_n = c$, then there exists some $N_2$ such that for all $n \geq N_2$ we have:
$c - \epsilon < x_n$
Let $N = \max\{N_1, N_2\}$, then for all $n \geq N$, we get:
$c - \epsilon < x_n < c + \epsilon$
$|x_n - c| < \epsilon$
Your proof is correct. You might write down the definition of $\limsup$ before concluding that $x_n<c+\epsilon$ and $x_n>c-\epsilon$.
$$\limsup_{n\to\infty}x_n=c\iff\forall \epsilon>0\ \exists N_1\in\mathbb N\ \forall n\ge N_1, -\epsilon<\sup\{x_n,x_{n+1}\ldots\}-c<\epsilon$$
After this, you may safely conclude that $x_n<c+\epsilon$ for $n\ge N_1$