<s>[INST] How can you prove $1^3 + 2^3+\cdots+(n-1)^3 < \frac{n^4}{4} < 1^3 + 2^3 + \cdots + n^3$ by induction? Can you provide the steps and corresponding explanations to prove the following predicate by induction?
$$P(n) := 1^3 + 2^3+\cdots+(n-1)^3 < \frac{n^4}{4} < 1^3 + 2^3 + \cdots + n^3$$
I've done some work on it myself by attempting to show that $\frac{k^4}{4} < \frac{(k + 1)^4}{4}$ for the RHS, but I don't understand exactly what I am doing.
Thank you.
Notice: This is not a homework question. I'm attempting to self-study Calculus over the Summer.
 [/INST] HINT
By induction we have


*

*Base case: $n=1\implies 0<\frac14<1$

*Induction step, we assume true


$$P(n) := 1^3 + 2^3+\cdots+(n-1)^3 < \frac{n^4}{4} < 1^3 + 2^3 + \cdots + n^3$$
and we need to prove
$$P(n+1) := 1^3 + 2^3+\cdots+(n-1)^3+n^3 < \frac{(n+1)^4}{4} < 1^3 + 2^3 + \cdots + n^3+(n+1)^3$$
then we have
$$1^3 + 2^3+\cdots+(n-1)^3+n^3 < n^3 +\frac{n^4}{4}$$
$$\frac{n^4}{4}+(n+1)^3< 1^3 + 2^3 + \cdots + n^3+(n+1)^3$$
then it suffices to prove that
$$n^3 +\frac{n^4}{4}< \frac{(n+1)^4}{4}<\frac{n^4}{4}+(n+1)^3$$
that is
$$n^3 < \frac{(n+1)^4}{4}-\frac{n^4}{4}<(n+1)^3$$
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