Show that $L^p + L^r$ with the norm $||f|| = \inf \{||g||_p + ||h||_r \ : \ f = g + h\}$ is a Banach space The following is exercise 6.4 in Folland's Real Analysis:

If $1 \leq p < r \leq \infty$, $L^p + L^r$ is a Banach space with norm $||f|| =\inf \{||g_p|| + ||h||_r \ : \ f = g + h\}$, and if $p < q < r$ the inclusion map $L^q \to L^p + L^r$ is continuous.

I think I was able to show the first part (being a Banach space), but I lack confidence, so I would appreciate if you could review the proof and point any flaws. Also, I would appreciate any hints in the direction of showing that the inclusion is continuous.
Thanks in advance and kind regards.
Attempt of solution:
First we show that $L^p + L^r$ is a vector space. Of course $0 \in L^p + L^r$. Let $f, g \in L^p + L^r$ and $\lambda \in \Bbb{C}$. Then for every $f_1, g_1 \in L^p$, $f_2, g_2 \in L^r$ such that 
$$
f = f_1 + f_2, \quad g = g_1 + g_2
$$
it holds that 
$$
f + \lambda g = (f_1 + g_1) + \lambda (f_2 + g_2) = h_1 + h_2
$$
with $h_1 \in L^p, h_2 \in L^r$. 
Now we show that $||\cdot||$ is a norm. Let $f, g \in L^p + L^r$. The triangle inequality is valid:
\begin{align*}
||f + g|| & = \inf\{||v||_p + ||w||_r \ : \ f + g = v + w\} \\
& = \inf \{||f_1 + g_2||_p + ||f_2 + g_2||_r \ : \ f = f_1 + f_2, g = g_1 + g_2\} \\
& \leq \inf\{||f_1||_p + ||g_1||_p + ||f_2||_r + ||g_2||_r \ : \ f = f_1 + f_2, g = g_1 + g_2\} \\
& \leq \inf\{||f_1||_p + ||f_2||_r \ : \ f = f_1 + f_2\} + \inf\{||g_1||_p + ||g_2||_r \ : \ g = g_1 + g_2\} \\
& = ||f|| + ||g||.
\end{align*}
Also, for $\lambda \in \Bbb{C}$
$$
||\lambda f|| = \inf\{||\lambda f_1||_p + ||\lambda f_2||_r \ : \ f = f_1 + f_2\} = |\lambda| \ ||f||.
$$
Lastly, if $||f|| = 0$ then $||f_1||_p = ||f_2||_r = 0$ for all $f_1 \in L^p, f_2 \in L^r$ such that $f = f_1 + f_2$. Then $f = 0 + 0 = 0$.
The space $L^p + L^r$ with the norm above is complete.  Let $\sum_1^\infty f_n$ be an absolutely convergent series in $L^p + L^r$, that is, $\sum_1^\infty ||f_n|| < \infty$. For each $n \in \Bbb{N}$ there exists $g_n \in L^p, h_n \in L^r$ such that $f_n = g_n + h_n$ and 
$$
||g_n||_p + ||h_n||_r < ||f_n|| + 2^{-n}.
$$
It follows that $\sum_1^\infty g_n$ and $\sum_1^\infty h_n$ are absolutely convergent series in $L^p$ and $L^r$, respectively, so they have limits $g \in L^p$ and $h \in L^r$. These series are also absolutely convergent in $L^p + L^r$ since, for example, 
$$
\Big| \Big|\sum_1^N g_n - g \Big|\Big| \leq \Big| \Big| \sum_1^N g_n - g \Big| \Big|_p + ||0||_r.
$$
Then $\sum_1^\infty f_n = \sum_1^\infty (g_n + h_n)$ has a limit $g + h$ in $L^p + L^r$, and therefore $L^p + L^r$ is a Banach space.
 A: For me everything was clear, except the last part. I didn't quite understand how you chose the sequences $\{f_n^1\}$ and $\{f_n^2\}$? Since in the above step $f_n^1$ and $f_n^2$ was an arbitrary decomposition of the function $f_n$ and for me it's not entirely clear how one should extract the subsequences such that they are Cauchy in $L^p$ and $L^r$ respectively. (It might also be that I didn't really understand your argument)
I believe the continuous embedding from $L^q$ into $L^p+L^r$ can be proven as follows:
Choose an arbitrary function $f$ in $L^q$, then we need to show that there is a constant $C>0$
\begin{align}
||f||_{L^p+L^r}\leq C ||f||_{L^q}.
\end{align}
The idea is that you can choose the decomposition into an $L^p$ function plus an $L^r$ function in such a way that you measure the "large part" of the function in $L^p$ and the "small part" of the function in $L^r$. Both parts can then be estimated by the $L^q$ norm of $f$. More precisely define (I don't know what your domain is, probably some subset of $\mathbb{R}^n$ or a general measure space, so I just call it $X$)
\begin{align}
A&=\{x\in X||f(x)|\geq1\}\\
B&=\{x\in X||f(x)|<1\}
\end{align}
If we denote the characteristic functions of $A$ by $\chi_A$ and of $B$ by $\chi_B$, then  $\chi_A+\chi_B=1$ and we can decompose $f$ as 
\begin{align}
f=f(\chi_A+\chi_B)=f\cdot\chi_A+f\cdot\chi_B
\end{align}
One can then check that $f\cdot\chi_A\in L^p$ and $f\cdot\chi_B\in L^r$ and we get
\begin{align}
||f||_{L^p+L^r}\leq ||f\chi_A||_{L^p}+||f\chi_B||_{L^r}
\end{align}
Let's check for the first term that it can be estimated by the $L^q$ norm of $f$, for the second term the argument is similar. 
For all $x\in A$ we have $|f(x)|\geq 1$. Since $p<q$ this implies $|f(x)\cdot \chi_A(x)|^p\leq |f(x)\cdot \chi_A(x)|^q$ for all $x\in X$ and therefore
\begin{align}
||f\chi_A||_{L^p}\leq ||f\chi_A||_{L^q}\leq ||f||_{L^q}
\end{align}
Analogously one can show $||f\chi_B||_{L^r}\leq ||f||_{L^q}$
and therefore
\begin{equation}
||f||_{L^p+L^r}\leq ||f\chi_A||_{L^p}+||f\chi_B||_{L^r}\leq 2||f||_{L^q}
\end{equation}
Hope this helps! 
