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How do I prove that $a_n$ and $b_n$ converges if and only if $a_n - b_n$ and $a_n + b_n$ converges?

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    The first step is trying. Have you? Is there one of the two directions that seems easier to you? – 2017-01-25

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($\Rightarrow$). Let $a_n$ and $b_n$ be two convergent sequences. Then, $\lim_{n \to \infty}a_n = L$ and $\lim_{n \to \infty}b_n = M$. Now consider the sequence $c_n = a_n + b_n$. Then, $$\lim_{n \to \infty} c_n = \lim_{n \to \infty}(a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty}b_n = L + M,$$ hence $c_n$ converges. Can you prove the other case yourself?

($\Leftarrow$). Given is that the sequences $a_n+b_n$ and $a_n-b_n$ converge. Then, \begin{align} \lim_{n \to \infty}a_n & = \lim_{n \to \infty} \frac{1}{2}(2a_n) \\ & = \lim_{n \to \infty} \frac{1}{2}(a_n + b_n + a_n - b_n) \\ & = \lim_{n \to \infty} \frac{1}{2}(a_n + b_n) + \lim_{n \to \infty} \frac{1}{2} (a_n - b_n) \\ & = \frac{1}{2}\lim_{n \to \infty}(a_n + b_n) + \frac{1}{2}\lim_{n \to \infty}(a_n - b_n) \\ & = \frac{1}{2}(L_1 + L_2), \end{align} so $a_n$ converges. Can you do the proof for $b_n$ yourself?

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    As writing goes, writing $\lim_{n\to\infty} a_n$ *presupposes* that the limit exists -- which you cannot in $\Leftarrow$. – 2017-01-25
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    If thats the case I could not have started out with $\lim_{n \to \infty} c_n$ at $\Rightarrow$ either right? – 2017-01-25
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    Indeed. ${}{}{}$ – 2017-01-25
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    For "$\Rightarrow$" $a_n-b_n$; is it as easy as $\lim_{n \to \infty} c_n = \lim_{n \to \infty}(a_n - b_n) = \lim_{n \to \infty} a_n - \lim_{n \to \infty}b_n = L - M,$ or am I mistaken here? – 2017-01-25
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    you're right. $\phantom{}$ – 2017-01-25