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I was doing a question on analysis which asked me to state if certain statements were true or false. One of the statements were:

"If $a_n > 0$ for all natural numbers $n$ and $a_n \to a$, then $a > 0$."

I thought that this would be true, but I was wrong - the mark scheme said it was false. Why is this so? Surely if every term in the sequence is positive, the limit must also be positive.

Thanks in advance!

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    Think about $a_n=1/n$,what is $a$ here? – 2017-01-10
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    @kingW3 Ah! Then the limit of an is 0 even though an > 0 for all n, which means the statement is false. Thanks! – 2017-01-10
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    Related: http://math.stackexchange.com/questions/1424273/let-a-n-be-a-convergent-sequence-of-positive-real-numbers-why-is-the-limit – 2017-01-10

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It is false. Sequences like $1/n$ or $1/2^n$ coverge to $0$.

In general we can say that if $s