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TOC Pumping Lemma

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In pumping lemma it says if there is a string whose length is >=N,where N is number of states in dfa,then language of machine is infinite,but there is one upper constraing also.

2N-1>=|W|>=N

Can anyone tell me about the use of 2N-1 here ?
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Use the Pumping Lemma to show that the following languages over Σ={�,�}Σ={a,b} are not regular. In each case, carefully describe the string that will be pumped and explain why pumping it leads to a contradiction. {aaabnan∣n≥0} {ww∣w∈Σ∗}
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