Show that if xn is a bounded sequence, then xn converges if and only if lim sup(xn) = lim inf (xn).

I know that limit inferior describes the smallest number, v, that a subsequence can converge to, and I know that limit superior describes the largest number, u, that a subsequence can converge to.

So if lim inf = lim sup, then all subsequences must converge to the same number, and if all convergent subsequences converge to a limit, L, then the sequence itself converges to a limit L. Is this correct thinking?

For the converse, would I say that if a sequence converges to a limit, L, then any subsequence converges to L, and by proxy, if all subsequences converge to L, lim sup = lim inf?

Looks good. For the other direction, you might bring into play the fact that there are subsequences of $\{x_n\}$ that converge to $\limsup x_n$ and $\liminf x_n$ respectively. You also don't need the boundedness hypothesis if you define convergence of a sequence $\{x_n\}$ to $\pm\infty$ in the natural way.