Prove that $\lim_{n\to\infty} |a_n| = |a|$ Let $\lim_{n\to\infty} a_n = a$. Prove that $\lim_{n\to\infty} |a_n| = |a|$.
First of all, I wonder about what does it mean the statement. I am just studying the theory today. I actually know that $a_n$ is the general term of a sequence (and of a series). i.e. it should equal the formula which gives the pattern of the sequence. If we plug a number in the n of the formula, we get the output for the number plugged in. Instead, does $a$ stand for the cluster point of a sequence, right?
Could you show me the proof of the statement above and tell me if I misunderstood something in my wording, please?
 A: We don't need to know the exact values of a sequence in order to talk about it.  To say that there is a sequence {$a_n$} means nothing more than we have a countable set of values in an order.  It doesn't mean anything about whether we have a definition for them. 
In practical use we often (but far from always) do have a definition for otherwise we may have no method of predicting or describing the sequence.  But there's no reason in to assume that in talking about a sequence we can predict or describe it.
In this case the only thing we know about this sequence is that it converges to a real number we have labeled $a$.  The definition of a limit explains what that means.  (For every $\epsilon$ > 0 then there is a large M such that for all   $n> M$ then the $a_n$ terms are less than $\epsilon$ away from a; i.e |$a_n - a| < \epsilon$.)  You need to show by this definition, that the limit of the absolute values of this sequence (whatever the **** the sequence actually is) that the absolute values converge to the absolute value of the original limit.
HINT:  if you follow the definition of converges precisely not hard.  If |a - b| < c, then what can be said about ||a| - |b||?
A: Suppose that $lim_{n{\rightarrow}{\infty}} a_n=a$
Then for each $\epsilon>0$ , there is $N \in \mathbb N$ such that for each $n>N$ , $|a_n-a|<\epsilon$.
By triangle inequality , we have $|a+b|\leq |a|+|b| \Rightarrow ||a|-|b||\leq |a-b|.$
So observe that $||a_n|-|a||\leq|a_n-a|$
Since $|a_n-a|<\epsilon \Rightarrow ||a_n|-|a||< \epsilon$
Therefore $lim_{n{\rightarrow}{\infty}} |a_n|=|a|$ for each $n>N$.
