Is it true that $\|f \|_{L^{p-1} }\leq \|f\|_{L^{p}}$? Is this true?
$$
\|f \|_{L^{p-1} }\leq \|f\|_{L^{p}}\;\;
$$  
Specifically I know  $\;\;\|f\|_{L^{2}} \leq \|f\|_{L^{\infty}}$ $\;$ but I can't figure out why?
 A: To prove the stated inequality (for normalized measure spaces): Let $X$ be a finite measure space and $q \le p$. We have using Hölder for $f \in L^p$
\begin{align*}
   \|f\|_q &= \bigl\||f|^q\bigr\|_1^{1/q}\\\
           &= \bigl\|1 \cdot |f|^q\bigr\|_1^{1/q}\\\
           &= \|1\|_{p/(p-q)} \bigl\||f|^q\bigr\|_{p/q}^{1/q}\\\
           &= \mu(X)^{(p-q)/p} \|f\|_p
\end{align*}
If $\mu$ is normalized, i. e. $\mu(X) = 1$ (for example in $[0,1]$ or $\mathbb T$ with normalized arclength), then $\|f\|_q \le \|f\|_p$ for every $q \le p$. especially $\|f\|_{p-1} \le \|f\|_p$.
A: Your first inequality is not necessarily true. With $X=[0,1]$, $p=2$ and $f(x)=x$ we have $$ \| f \|_1 = \int^1_0 x dx = 1/2 $$ while $$ \| f \|_2 = \left( \int^1_0 x^2 dx \right)^{1/2}=1/9 .$$
However your second one is correct: $$ \| f \|_p \leq \| f \|_{\infty}. $$ This follows so quickly from the definition of essential supremum and basic estimates, I recommend you try to prove this again. 
Whilst we don't have you first inequality, we do have the follow inclusion: $ L^p \subseteq L^q $ for a finite measure space and $ p \leq q .$ This follows from an application of Holder's inequality. I suggest you give that exercise a try as well.
In general measure spaces however, there are not any guaranteed inclusions.  This great question shows that. 
