Proving a series of functions does not converge uniformly on $\mathbb{R}$ I'm working through a question (it's a question with parts that builds on each other) that overall will show: 


*

*a series of functions does not converge uniformly on $\mathbb{R}$ by
showing the sequence of its partial sums is not uniformly Cauchy on
$\mathbb{R}$


The series of functions is: $$\sum_{n=1}^{\infty}{\dfrac{1}{\sqrt{n}}\sin\left(\frac{x}n\right)} \quad  
$$
Firstly, I'm asked to show the negation for uniformly Cauchy:


*

*For some $\epsilon > 0$, for all $n$ $\epsilon$ $\mathbb{N}$, there
exists an $m, n \geq N$ and there exists an $x$ $\epsilon$ I such
that $|f_m{(x)} - f_n{(x)}| \geq \epsilon$ 


Secondly: If $k, N$ $\epsilon$ $\mathbb{N}$ and $N + 1 \leq k \leq 4N + 3$, show that :


*

*$1 \geq \sin\left(\frac{N+1}k\right) \geq \frac{1}5$


I have done this on paper, but would take while to write out, so will omit my answer here.
Third (where I'm beginning to get stuck): Show that for all $N$ $\in$ $\mathbb{N}$
$$\sum_{k=N+1}^{k=4N+3}{\dfrac{1}{\sqrt{k}}} \geq 2\sqrt{2}$$
My idea for part 3 is to compare it to an integral, specifically the integral of $\frac1{\sqrt{x}}$, but I'm not sure how to show this relation for all $N$ (what the limits of integration should be) - any tips?
Lastly:


*

*Show that when $x = N + 1, m = 4N + 3$, and $n = N$, $$|S_m{(x)} -
   S_n{(x)}| \geq \frac{2\sqrt{2}}5$$
And then ultimately show the original sequence doesn't uniformly
converge on $\mathbb{R}$
Part C I think I'm on the right track and some tips should suffice, but for part d I'm relatively lost. Any help's greatly appreciated - thank you
 A: Part 2
Let $N + 1 \le k \le 4N + 3$.  
Then $\frac{N+1}{k}$ satisfies the inequalities $\frac{N+1}{4N+3}\le\frac{N+1}{k}\le1$.  
Now, for $x$ in the interval $\left(\frac{N+1}{4N+3},1\right)$, $\sin x$ is monotonically increasing and we have  
$$1\ge \sin(1)\ge \sin\left(\frac{N+1}{k}\right)\ge\sin\left(\frac{N+1}{4N+3}\right)$$
and thus
$$\begin{align}
\sin\left(\frac{N+1}{k}\right)&\ge\sin\left(\frac{N+1}{4N+3}\right)\\\\
&\ge\sin(1/4)\\\\
&\ge \frac14-\frac{1}{3!}(\frac14)^3\\\\
&=\frac14\left(1-\frac{1}{96}\right)\\\\
&>\frac15
\end{align}$$
which was to be shown.

Part 3
For this part, we will use the result from the following integral
$$\begin{align}
\int_{N+1}^{4N+4}\frac{dx}{\sqrt{x}}&=2\left(\sqrt{4N+4}-\sqrt{N+1}\right)\\
&=2\sqrt{N+1}(2-1)\\
&\ge 2\sqrt{2}
\end{align}$$
for $N\ge1$.
Now, we note that this integral can be represented by the following summation:
$$\begin{align}
\int_{N+1}^{4N+4}\frac{dx}{\sqrt{x}}&=\sum_{k=N+1}^{4N+3}\int_{k}^{k+1} \frac{dx}{\sqrt{x}}\\
&\le\sum_{k=N+1}^{4N+3} \frac{1}{\sqrt{k}}
&\ge 2\sqrt{2}
\end{align}$$
and we have the desired inequality!

Part 4
From the Part 2, we showed that if $N + 1 \le k \le 4N + 3$, then
$$1 \ge \sin\left(\frac{N+1}{k}\right) \ge \frac15$$
From the Part 3, we showed that for all $N$
$$\sum_{k=N+1}^{4N+3}{\frac{1}{\sqrt{k}}} \ge 2\sqrt{2}$$
Then, putting these together we have 
$$\begin{align}
|S_m(x)-S_n(x)|&=\left|\sum_{k=N+1}^{4N+3} \frac{\sin((N+1)/k)}{\sqrt{k}}\right|\\
&\ge\left|\sum_{k=N+1}^{4N+3} \frac{\frac15}{\sqrt{k}}\right|\\\\
&\ge\frac15\left|\sum_{k=N+1}^{4N+3} \frac{1}{\sqrt{k}}\right|\\\\
&\ge\frac{2\sqrt{2}}{5}
\end{align}$$
