weak convergence of a sequence of functions in $L^p(\mathbb{R})$ I have a problem and would like a hint or two if possible. 

Problem: Let $f_0(x)\in L^P(\mathbb{R})$ for $1<p<\infty$ and for each natural number $n$ define $f_n(x)=f_0(x-n)$ for all $x\in\mathbb{R}$. Show that $\{f_n\}\rightharpoonup f$ where $f\equiv 0$ on $\mathbb{R}$. Is this true for $p=1$? 

My thoughts so far: 
I had a couple of ideas, most turned out to be fruitless; however, my current idea is that I might be able to show that $f_n\rightarrow f$ point wise almost everywhere on $\mathbb{R}$ which would give me weak convergence (I have a theorem in my book to help with this part). 
Only problem is that I'm not sure how to show that. It would make sense that I can somehow use the fact that each $|f_n|^p$ is integrable over $\mathbb{R}$ but I'm not certain where to go from there. 
Hopefully my thought is in the right direction. Thank you for you help/advice! :) 
P.S. I'm using Royden Fitzpatrick 4th edition. Question is number 15 from chapter 8. 
 A: Let $1 < p < \infty$ be as stated. You are trying to show that $\lim_{n} f_{n}g\,dt = 0$ for each $g \in L^{q}(\mathbb{R})$, where $\frac{1}{p}+\frac{1}{q}=1$. Let $\epsilon > 0$ be given. For any $g \in L^{q}(\mathbb{R})$ for $1 < q < \infty$, one has
$$
               L^{q}-\lim_{N}\chi_{[-N,N]}g=g.
$$
Therefore, for any $\epsilon > 0$, there exists $N$ large enough that
$$
                 \|f\|_{p}\|g-\chi_{[-N,N]}g\|_{q} < \epsilon/2.
$$
And there exists $K$ large enough that $\|f-\chi_{[-K,K]}f\|_{p}\|g\|_{q} < \epsilon/2$. Then write
$$
        \int f_{n}g\,dt = \int f_{n}(g-\chi_{[-N,N]}g)\,dt+\int f_{n}\chi_{[-N,N]}g\,dt\\
       = \int f_{n}(g-\chi_{[-N,N]}g)\,dt +\int_{n-N}^{n+N}f(x)\chi_{[-N,N]}(t-n)g(t-n)\,dt.
$$
If $n \ge K+N$, then $[-n+N,n+N]\subset [K,\infty)$, which gives the bound
$$
              \left|\int f_{n}g\,dt\right| \le \|f\|_{p}\|g-\chi_{[-N,N]}g\|_{q}+\|f-\chi_{[-K,K]}f\|_{p}\|g\|_{q} < \epsilon,\;\;\; n \ge N+K.
$$
Therefore, if $f \in L^{p}$ for some $1 < p < \infty$, then $\{ f_{n}\}$ converges weakly to $0$. This is not true for $p=1$; to see this, let $f=\chi_{[0,1]}$ and let $g$ be the constant function $1$ on $\mathbb{R}$. Then $g \in L^{\infty}$ and $\int f_{n}g\,dt = 1$ for all $n =1,2,3,\cdots$. 
A: This answer is tailored for readers of the textbook to which TK-421 refers.
This generalizes the example spanning pages 167–168. Let $A$ be any measurable subset of $\Bbb R$ of finite measure. Because $A$ has finite measure, it is contained in some interval $[a,b]$. Next, consider sequence $\bigl\{f_0\chi_{[-n,n]}\bigr\}$, which clearly is in $L^p(\Bbb R)$ and converges pointwise on $\Bbb R$ to $f_0$. Moreover,
$$\lim_{n\to\infty}\int_\Bbb R\bigl\lvert f_0\chi_{[-n,n]}\bigr\rvert^p =\lim_{n\to\infty}\int_{-n}^n\lvert f_0\rvert^p =\int_\Bbb R\lvert f_0\rvert^p.$$
Hence, $\bigl\{f_0\chi_{[-n,n]}\bigr\}\to f_0$ in $L^p(\Bbb R)$ by Theorem 7 of Section 7.3. Then for every $\epsilon > 0$, there is an index $N > b - a$ for which $\bigl\lVert f_0 - f_0\chi_{[-N,N]}\bigr\rVert_p <\epsilon\big/m(A)^{1/q}$. That shows that if $A$ is outside of $[-N,N]$, then the norm on $L^p(A)$ of $f_0$ will be less than $\epsilon\big/m(A)^{1/q}$. Now $f_0(x - N)$ shifts $f_0(x)$ far enough to the right along the $x$-axis so that $A$ is outside of $[-N,N]$ with respect to $f_0(x)$ so that on $A$, $\lVert f_0(x - N)\rVert_p <\epsilon\big/m(A)^{1/q}$. Next, define $g\equiv 1$ on $A$. Function $g$ clearly belongs to $L^q(A)$, and $\lVert g\rVert_q =\bigl(\int_A\lvert g\rvert^q\bigr)^{1/q} =\bigl(\int_A 1^q\bigr)^{1/q} = m(A)^{1/q}$. Then using Hölder's inequality,\begin{align}
\lim_{n\to\infty}\int_A f_n &\le\lim_{n\to\infty}\int_A\lvert g(x)f_0(x - n)\rvert\,dx\\
&\le\lim_{n\to\infty}\lVert g\rVert_q\lVert f_0(x - n)\rVert_p\\
& < m(A)^{1/q}\cdot\epsilon\big/m(A)^{1/q} =\epsilon.\end{align}
Because that holds for all $\epsilon$, it holds also for $\epsilon = 0$. We thereby infer from Theorem 10 that $\{f_n\}\rightharpoonup f$ in $L^p(\Bbb R)$.
That is not true for $p = 1$ by the counterexample in the example spanning pages 167–168: $\int_\Bbb R f_n = 1$ for all $n$, but $\int_\Bbb R f = 0$.
A: Fix $g\in L^q$, the dual space. Given $\epsilon>0$, there is $M$ such that both $f \chi_{|x|\ge M}$ and $g \chi_{|x|\ge M}$ have norms (in their respective spaces) less than $\epsilon$. Use Hölder's inequality to show that for $n>2M$,
$$\int_{\mathbb R} |f(x-n)g(x)|\,dx \le \epsilon  \|f\|_{L^p}+\epsilon \|g\|_{L^q} $$

Hint for $p=1$: the answer is negative, and the reason is that an $L^\infty$ function
does not need to have a "small tail" at infinity.
