LIMIT SUPERIOR, LIMIT INFERIOR

LIMIT SUPERIOR, LIMIT INFERIOR

SEQUENCES

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A number l is called the limit superior, greatest limit or upper limit (lim sup or lim) of the sequence

fu_ng if in?nitely many terms of the sequence are greater than l                    while only a ?nite number of terms are

greater than l ‏         , where      is any positive number.

A number l is called the limit inferior, least limit or lower limit (lim inf or lim) of the sequence fu_ng if

in?ntely many terms of the sequence are less than l ‏               while only a ?nite number of terms are less than

l       , where      is any positive number.

These correspond to least and greatest limiting points of general sets of numbers.

If in?ntely many terms of        fu_ng  exceed any positive number        M, we de?ne lim sup fu_ng ¼ 1.              If

in?nitely many terms are less than            M, where M is any positive number, we de?ne lim inf fu_ng ¼ 1.

If lim u_n¼ 1, we de?ne lim sup fu_ng ¼ lim inf fu_ng ¼ 1.

n!1

If lim u_n¼ 1, we de?ne lim sup fu_ng ¼ lim inf fu_ng ¼ 1.

n!1

Although every bounded sequence is not necessarily convergent, it always has a ?nite lim sup and

lim inf.

A sequence fu_ng converges if and only if lim sup u_n¼ lim inf u_nis ?nite.

NESTED INTERVALS

Consider a set of intervals ½a_n; b_n, n ¼ 1; 2; 3; . . . ; where each interval is contained in the preceding

one and lim ًa_nb_nق ¼ 0.                Such intervals are called nested intervals.