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Math Help - limit superior inequality

  1. #1
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    limit superior inequality

    Let an, bn be two bounded sequences. Show that

    limn-->∞ sup (an+bn) ≤ limn-->∞ sup an + limn-->∞ sup bn
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  2. #2
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    That's because \sup_{m\ge n}(a_m+b_m)\le(\sup_{m\ge n}a_m)+(\sup_{m\ge n}b_m). Indeed, on the right you can, roughly speaking, choose the largest elements of the sequences \{a_m\} and \{b_m\} independently and then add them, whereas on the left you have to add elements with the same index and then choose the largest.
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