= Solution
The relevant <Banach-Alaoglu theorem> says that for $1<p<\infty$ the closed bounded balls of $L^p(\mathbb R^n)$ are compact in the <weak topology>. In particular, every bounded sequence has a weakly convergent subsequence whose limit has <norm> no larger than the common bound. Applying the preceding <duality of Lp spaces> also identifies this as weak-star compactness of $L^p=(L^q)^*$; its weak and weak-star test spaces are both $L^q$.
Here is a direct proof of both compactness and the subsequence assertion. Let $\|f_j\|_p\leq M$ and choose a countable dense set $\{g_m\}$ in the separable <Lp space> $L^q$. Each scalar sequence $\int f_jg_m$ is bounded. Successive subsequence selection followed by the <diagonal subsequence argument> gives $f_{j_k}$ for which these pairings converge for every $m$. For arbitrary $g\in L^q$,
$$
\left|\int(f_{j_k}-f_{j_l})g\right|\leq2M\|g-g_m\|_q+\left|\int(f_{j_k}-f_{j_l})g_m\right|.
$$
First approximate $g$ by $g_m$ and then take $k,l$ large. This proves convergence of every pairing, and the limit $\Lambda(g)=\lim_k\int f_{j_k}g$ is linear with $|\Lambda(g)|\leq M\|g\|_q$. Apply the already proved <duality of Lp spaces> with $q$ in place of $p$: $\Lambda(g)=\int fg$ for an $f\in L^p$ with $\|f\|_p\leq M$. Thus
$$
\boxed{f_{j_k}\rightharpoonup f\text{ in }L^p,\qquad \|f\|_p\leq M.}
$$
To obtain compactness rather than only a sequential statement, on the radius-$M$ ball define
$$
d(f,h)=\sum_{m=1}^{\infty}2^{-m}\frac{|\int(f-h)g_m|}{1+|\int(f-h)g_m|}.
$$
Density and <Hölder's inequality> show that this is a <metric space> distance and that its topology is exactly the <weak topology> on this ball: every remaining test pairing is uniformly approximated there by a dense-set pairing. The subsequence argument proves sequential compactness for this metric. A sequentially compact <metric space> is compact: otherwise either an infinite separated sequence contradicts <total boundedness> or a Cauchy sequence without a limit contradicts sequential compactness; completeness and <total boundedness> give compactness by successive finite coverings. This completes the proof. The same argument is the <Sequential Banach-Alaoglu theorem for a separable predual>.
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