Task: Show the convergence of sequence $\{a^k\}$ in $\mathbb{R}^3$, where $$a_k=\begin{pmatrix}a^k_1\\a_2^k\\a^3_k\end{pmatrix}\in\mathbb{R}^3$$ with $a_1^k=\left(\frac{k}{k+2}\right)^k$, $a_2^k=\left(k-\sqrt{k^2-1}\right)^k$, $a_3^k=k\sin\frac{1}{k}$.
Question: Do I just need to show that every $a_i^k$ for $i\in\{1,2,3\}$ is convergent in order to show that $a_k$ is convergent? I would just compute the limits like that: \begin{align} a_1=\left(\frac{k}{k+2}\right)^k=\left(\frac{1}{1+\frac{2}{k}}\right)^k=\frac{1}{\left(1+\frac{2}{k}\right)^k}\overbrace{\to}^{k\to\infty}e^{-2}\in\mathbb{R} \end{align}
etc. and then I would come to the conclusion that all of the $a_i$'s are convergent so is $a_k$. Is this correct?
Edit: How can I calculate $\lim_{k\to\infty}a_2^k$? I've got to $$\exp\left\{\lim_{k\to\infty}k\cdot \lim_{k\to\infty} \ln \frac{1}{k+\sqrt{k^2-1}}\right\}$$ but my solution is, that this equals $e^0=1$ which is not true, according to Wolfram Alpha. Can you help me?