<s>[INST] Prove that $1^3 + 2^3 + \cdots + n^3 < n^4$. I am trying to prove the following: $1^3 + 2^3 + \cdots + n^3 < n^4$ if $n \in \mathbb{N}, n>1$ by induction. From there, I am to prove that the sum is $< \frac{n^4}{2}$ if $n>2$. My attempt to do so is as follows:
First, we use $n = 2$ as a basis step since $n \in \mathbb{N}, n>1$ which shows that $1^3+2^3 < 2^4 \Rightarrow 9<16$ which is true. Now, for $n+1$, we have $n^3 + (n+1)^3 < n^4 + (n+1)^4$ by the induction hypothesis. By expanding the two binomials, we have
$2n^3+3n^2+3n+1< 2n^4+4n^3+6n^2+4n+1$. By simplifying our expression, we are left with
$0<n(2n^3+2n^2+3n+1)$ which would always be true with $n \in \mathbb{N}, n>1$. 
I am not sure if this proof would be valid, or if it is logically correct. As for the second part of the question, I would not know where to begin. Could somebody please provide a hint or point out where this goes awry? Any suggestions and/or criticism is welcome. 
I am using the textbook Introduction to Analysis by Arthur Mattuck. 
 [/INST] As the others have said, you went wrong in the induction step. Here's one way to prove what we want without having to expand the binomials:
\begin{align*}
1^3 + 2^3 + \ldots + (n+1)^3 &= \left[1^3 + 2^3 + \ldots + n^3 \right] + (n+1)^3 \\
&< n^4 + (n+1)^3 \qquad\text{by the induction hypothesis} \\
&= n(n)^3 + 1(n+1)^3 \\
&< \color{red}{n}\color{blue}{(n+1)^3} \color{red}{+ 1}\color{blue}{(n+1)^3} \\
&= (\color{red}{n+1})\color{blue}{(n+1)^3}  \qquad\text{factor out the common }(n+1)^3 \text{ term} \\
&= (n+1)^4 \\
\end{align*}
as desired.
</s>