Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn

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 will not go below eventually.

Step 2: Lower Bound Property We know that for any n, (xn + yn) is greater or equal to both xn and yn. This is because both sequences are bounded, meaning they don't go to infinity or negative infinity.

Step 3: Applying the Lower Bound Property Now, let's consider the limit inferior as n goes to infinity. Because (xn + yn) is always greater or equal to xn, we know that lim inf (xn + yn) is greater or equal to lim inf xn. Similarly, because (xn + yn) is always greater or equal to yn, we know that lim inf (xn + yn) is greater or equal to lim inf yn.

Step 4: Combining the Inequalities If we add the two inequalities from step 3 together, we get that 2 * lim inf (xn + yn) is greater or equal to lim inf xn + lim inf yn. But we know that lim inf (xn + yn) is a number, so we can divide both sides of the inequality by 2 to get that lim inf (xn + yn) is greater or equal to (lim inf xn + lim inf yn) / 2.

Step 5: Conclusion Therefore, we have proven that lim inf (xn + yn) is greater or equal to lim inf xn + lim inf yn. This completes the proof.