<s>[INST] Mathematical Induction Inequality problem I am trying to solve the following problem with  mathematical induction:
$$
\forall n>1,\qquad \frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<\frac{n-1}{n}
$$
but since I am new to the concept when it comes to inequalities I can't quite seem to work it out. 
Help, anyone?
 [/INST] Let $$f(n) = \frac{1}{2^2} + \cdots + \frac{1}{n^2}.$$ Now we want to show that $$f(n)<\frac{n-1}{n}\tag{1}$$ for all integers greater than $1$. A proof by induction consists of two equally important steps. In the base case we show that $(1)$ indeed holds for $n=2$. In the inductive step we assume that $(1)$ is true for some number $n$ and use that to show that it is also true for $n+1$. 
Base case: We have $f(2)=\frac{1}{4} < \frac{1}{2}=\frac{2-1}{2}.$
Inductive step: Assume $(1)$ is true for some $n \geq 2$. We can calculate 
\begin{align*}
f(n+1) &= \color{green}{f(n)} + \frac{1}{(n+1)^2} \\
& < \color{green}{\frac{n-1}{n}} + \frac{1}{(n+1)^2} \\
& < \frac{n-1}{n} + \frac{1}{n\cdot(n+1)} \\
&= \frac{(n+1)-1}{n+1}.
\end{align*}
The expressions in green indicate the essential part of the inductive step. This shows that $(1)$ is true for $n+1$. Together with the base case this proves $(1)$ for all $n\geq 2$.
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