Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn
Question
Let and be bounded sequence. Prove that ( \liminf_{n \to \infty} (x_n + y_n) \geq \liminf_{n \to \infty} x_n + \liminf_{n \to \infty} y_n. \
Solution
Sure, here is the proof:
Step 1: Definitions First, let's define what we mean by lim inf. The limit inferior (or infimum limit) of a sequence (xn) is the greatest lower bound of the set of all limit points of the sequence. In other words, it's the largest value that we can guarantee the sequence wi
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