<s>[INST] Proof Verification of a Limit in Real Analysis Is this following proof correct?
The limit
\begin{align*}
\lim_{n\to\infty}\frac{t^{n+1}}{(n+1)!}
\end{align*}
tends to zero if for every $\epsilon>0$, there exists $N_\epsilon$ such that 
\begin{align*}
n\geq N_\epsilon\implies \left|\frac{t^{n+1}}{(n+1)!}\right|<\epsilon
\end{align*}
For any $t\in\mathbb{R}$, there exists $N\in\mathbb{N}$ such that $t<N$. Let $N$ be the least such natural number. Then
\begin{align*}
\frac{t^{N+k}}{(N+k)!}&=\frac{t\cdot t\cdots t\cdot t\cdots t\cdot t}{1\cdot 2\cdots (N-1)N\cdots(N+k-1)(N+k)}\\
&<\frac{t\cdot t\cdots t \cdot t\cdots t\cdot t}{1\cdot 1\cdots 1\cdot N\cdots N\cdot N}\\
&=t^N\left(\frac{t}{N}\right)^{k}
\end{align*}
Let $a=\frac{t}{N}$. Since $t<N$, $a=\frac{t}{N}<1$. If
\begin{align*}
\frac{t^{N+k}}{(N+k)!}<t^N\left(\frac{t}{N}\right)^{k}=t^Na^{k}<\epsilon
\end{align*}
then dividing both sides by $t^N$,
\begin{align*}
a^{k}<\frac{\epsilon}{t^N}
\end{align*}
Taking $\log_a$ of both sides reverses the inequality because it is a decreasing function. Hence
\begin{align*}
k>\log_{a}\left(\frac{\epsilon}{t^N}\right). 
\end{align*}
Any value for $k$ greater than this expression will ensure the limit remains within the desired bounds. Therefore, take $N_\epsilon=N+k$. Since the choice of $t$ was arbitrary, we may conclude that the limit converges throughout the entirety of $\mathbb{R}$.
 [/INST] The proof looks good (actually very good) to me.  The only improvement I can suggest is to turn it around a little, so that no cleverness is invoked.
Proof: Given $t>0$ and $\epsilon > 0$ arbitrary, put:
$$\begin{align*}
  N &= \lfloor t \rfloor + 1  & (\text{$N$ is the smallest natural number strictly greater than 1})
\\
  a &= \tfrac{t}{N}           & (\text{so $a < 1$})
\\
  k &= \left\lceil \log_{a}\left( \frac{\epsilon}{t^N} \right) \right\rceil
\\
  N_{\epsilon} &= N+k
\end{align*}$$
Then if $n > N_{\epsilon}$ we have
$$\begin{align*}
  \frac{t^{n+1}}{(n+1)!} &= \frac{t^{N+n-N+1}}{N!(N+1)(N+2)\dots(n+1)}
\\
  &<
  \frac{t^N}{1^N} \cdot \left( \frac{t}{N} \right)^{n+1-N}
\\
  &=
  t^N a^{n - N_{\epsilon} + N_{\epsilon} +1-N}
\\
  &<
  t^N a^{k+1} & (\text{via $N_{\epsilon} = N+k$, $n > N_{\epsilon}, a < 1$)}
\\
  &<
  t^N \frac{\epsilon}{t^N}  & (\text{via construction of $k$})
\\ 
  &=
  \epsilon
\end{align*}$$
as required.
Remark: Your calculations did all the hard work of finding $N_{\epsilon}$.  All this answer suggests is to summarize it into easier(?) steps from 'if $n > N_{\epsilon}$' to 'then $(\ldots) < \epsilon$'.
</s>