<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: For the left hand side inequality we have
$$ 1^3 + 2^3 +\ldots +n^3 < \frac{n^4}4 + n^3 < \frac{n^4}4 + \frac14 (4n^3 + 6n^2 + 4n + 1) = \frac{(n+1)^4}{4} $$
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