Suppose $\{f_n\}$ is a sequence in $L^p$ such that for each $g\in L^q$, the sequence $\{\int f_n g\}$ is bounded. Then $\{f_n\}$ is bounded in $L^p$. $(1\leq p<\infty)$
Proof: Argue by contradiction, suppose there exists a subsequence such that $$||f_{n_k}||_p \geq k.$$
Then we define a sequence $g_k \in L^q$ (this is the conjugate function from Royden's book), $$g_k = ||f_{n_k}||_p^{1-p} \operatorname{sgn}(f(x)) |f(x)|^{p-1},$$ we have that $||g_k||_q = 1$ and $\int f_{n_k} g_k = ||f_{n_k}||_p \geq k$ which contradicts the assumption.
Is this okay? thank you very much!
As pointed out in the comments, this is easier to prove if you use the uniform boundedness principle. I will assume you meant that $\frac1q+\frac1p=1$.
Now, since $(L^q(\mathbb{R}))'$ is isometric to $L^p(\mathbb{R})$, we can interpret the sequence $(f_n)$ as a sequence of functionals in $(L^q(\mathbb{R}))'$. Then the condition says that for each $g\in L^q(\mathbb{R})$, we have that $\sup_{n\in \mathbb{N}} |f_n(g)|<\infty$. Since $L^q(\mathbb{R})$ is a Banach space, the uniform boundedness principle implies $\sup_{n\in \mathbb{N}}\|f_n\|<\infty$.