# show by induction that the sequence is monotonically increasing

show by induction that the sequence : is monotonically increasing ... any help? it's a bit hard for me to write the induction hypothesis, I know that we must show that :

        $$a_1\le a_2$$
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Let it be for every natural n:
$$a_n\le a_{n+1}$$ ----> hypothesis
and we must prove that it is also correct for $$n+2$$:
$$a_{n+1}\le a_{n+2}$$


First what I said is correct? and I don't know how to continue from here... Any help is appreciated.

We don't need to use induction, we just need to show $$a_n < a_{n+1}$$.
Let's start with $$\begin{eqnarray*} 2 &<& 3 \\ 4n+2 &<& 4n+3 \\ 2(2n+1) &<& 2n+2 + 2n+1 \end{eqnarray*}$$ divide by $$2(n+1)(2n+1)$$ and we have $$\begin{eqnarray*} \frac{1}{n+1} < \frac{1}{2n+1} + \frac{1}{2(n+1)} \end{eqnarray*}$$ add $$\frac{1}{n+2} +\cdots + \frac{1}{2n}$$ to both sides, so $$\begin{eqnarray*} \frac{1}{n+1} +\cdots + \frac{1}{2n} < \frac{1}{n+2} + \cdots + \frac{1}{2(n+1)}. \end{eqnarray*}$$ Which means $$a_n .