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Upper and Lower Linits (lim sup and lim inf) - Sohrab Proposition 2.2.39 (b) ....
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[QUOTE="StoneTemplePython, post: 6776463, member: 613025"] I presume you know monotone convergence theorem... $u_n$ is a monotone non-decreasing sequence and the sequence is bounded, so it has a limit $L_u$ $v_n$ is a monotone non-increasing sequence and is bounded so it has a limit $L_v$ but for all $n$ $u_n \leq v_n$ by construction hence passing limits, $L_u \leq L_v$ (This is another basic property of limits... if you aren't familiar, argue by contradiction that $L_u - L_v = c \gt 0$, now select something easy, say $\epsilon := \frac{c}{10}$ which implies there is some $N$ such that for all $n\geq N$ in each sequence (a slightly more careful approach is $N =\max\big(N_v, N_u\big)$) you have ==== [S]$\vert u_n - L_v\vert \lt \epsilon$ [/S] [B]edit:[/B] to cleanup a typo, this should have said: $\vert u_n - L_u\vert \lt \epsilon$ . I had the wrong subscript. ==== and $\vert v_n - L_v\vert \lt \epsilon$ but this implies $u_n \gt v_n$ which is a contradiction -- sketching this out is best... it implies that $v_n \lt L_v + \frac{1}{10}c \lt L_v + \frac{9}{10}c = L_v + c - \frac{1}{10}c = L_u- \frac{1}{10}c \lt u_n$ ) [/QUOTE]
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Upper and Lower Linits (lim sup and lim inf) - Sohrab Proposition 2.2.39 (b) ....
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