Let (an)neN) CR be a sequence. (a) Show that lim supn an suPnen an. (b)...

  

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Let (an)neN) CR be a sequence. (a) Show that lim supn→ an ≤ suPnen an. (b) Find a sequence (an) for which the inequality above is strict. (c) Show that lim infn→∞ an ≤ lim supn→∞ an. (d) Show that the limits superior and inferior both exist in RU{x}. Hint: For the limit superior, consider separately the case where (an) is bounded versus unbounded. Consider the following functions sin , f(x) = { 0, (a) Sketch f(x) and g(x) for x = (-π, π). x 0 x = 0' }={800₂² g(x) = x sin, x0 0, x = 0 (b) Which of them are continuous at x = 0? Hint: For f, consider sequential continuity. For g recall sin y ≤ 1 for every y € R. | Let (an)neN) CR be a sequence. (a) Show that lim supn→ an ≤ suPnen an. (b) Find a sequence (an) for which the inequality above is strict. (c) Show that lim infn→∞ an ≤ lim supn→∞ an. (d) Show that the limits superior and inferior both exist in RU{x}. Hint: For the limit superior, consider separately the case where (an) is bounded versus unbounded. Consider the following functions sin , f(x) = { 0, (a) Sketch f(x) and g(x) for x = (-π, π). x 0 x = 0' }={800₂² g(x) = x sin, x0 0, x = 0 (b) Which of them are continuous at x = 0? Hint: For f, consider sequential continuity. For g recall sin y ≤ 1 for every y € R. |

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a Using the knowledge that lim sup a is the supremum of all subsequential limits of the sequence a w... View the full answer