Using the Uniform Continuity of the Characteristic Function to Show it's Differentiable I am working on part (iii) of exercise 3.3.17. in Durrett's Probability book. 
See this question: Relationship between the weak law of large numbers and characteristic functions
Basically, my issue is this. $\varphi$ is the ch.f. of a random variable. We know that $n(\varphi(t/n)-1)\to iat$ for all $t$ as $n\to\infty$ through the integers.  I am trying to show that 
$$
\frac{\varphi(h)-1}{h}\to ia
$$
as $h\downarrow 0$ in an arbitrary way, thereby proving that $\varphi'(0)$ exists. I know that $\varphi$ is uniformly continuous. Following the solution suggested in the above link, I considered the following approach. 
Let $h_{n}$ be an arbitrary sequence of real numbers descending to $0$. For an arbitrary $\delta >0$, we may write $h_{n}=t_{n}/m_{n}$ for sufficiently large $n$, where $m_{n}\in\mathbb{N}$ and $|t_{n}-1|<\delta$. Hence, for sufficiently large $n$, 
$$
\begin{aligned}
\left|\frac{\varphi\left(\frac{t_{n}}{m_{n}}\right)-1}{\frac{t_{n}}{m_{n}}}-ia\right|&=\frac{1}{t_{n}}\left|m_{n}\left(\varphi\left(\frac{t_{n}}{m_{n}}\right)-1\right)-iat_{n}\right|\\
&\leq\frac{1}{1-\delta}\left|m_{n}\left(\varphi\left(\frac{t_{n}}{m_{n}}\right)-\varphi\left(\frac{1}{m_{n}}\right)\right)\right|\\
&\qquad+\frac{1}{1-\delta}\left|m_{n}\left(\varphi\left(\frac{1}{m_{n}}\right)-1\right)-ia\right|\\
&\qquad+\frac{1}{1-\delta}\left|ia(1-t_{n})\right|
\end{aligned}
$$
Question:
The last two terms I can control, my question is can I use the uniform continuity of $\varphi$ to prove that 
$$
\left|m_{n}\left(\varphi\left(\frac{t_{n}}{m_{n}}\right)-\varphi\left(\frac{1}{m_{n}}\right)\right)\right|\to 0\qquad\text{ as }n\to\infty?
$$
 A: As @NateEldredge pointed out in his answer to the question you mentioned, you have to show that the convergence
$$\lim_{n \to \infty} n (\varphi(t/n)-1) = iat$$
holds locally uniformly (in $t$). Using the locally uniform convergence, it's not difficult to answer your question. Note that by the triangle inequality
$$\begin{align*} \left| m_n \left[ \varphi \left( \frac{t_n}{m_n} \right)- \varphi \left( \frac{1}{m_n} \right) \right] \right| &\leq 2 \sup_{t \in [1-\delta,1+\delta]} \left| m_n \left[ \varphi \left( \frac{t}{m_n} \right)-1 \right] -iat \right|+|ia(t_n-1)| \\ &\leq 2 \sup_{t \in [1-\delta,1+\delta]} \left| m_n \left[ \varphi \left( \frac{t}{m_n} \right)-1 \right] -iat \right| + |a| \, \delta . \end{align*}$$
Becuase of the locally uniform convergence, we know that the first term on the right-hand side converges to $0$ as $n \to \infty$ (for fixed $\delta>0$). Hence,
$$\limsup_{n \to \infty} \left| m_n \left[ \varphi \left( \frac{t_n}{m_n} \right)- \varphi \left( \frac{1}{m_n} \right) \right] \right| \leq |a| \, \delta.$$
Letting $\delta \to 0$ finishes the proof.
