Proving the limit laws

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Addition. Let be a subset of the real numbers, be real functions, and be real numbers. Now suppose and . We would like to show that .

Since and , we know that and Now let . We need to find such that But by the triangle inequality. So if we can show , then our result follows immediately. But, being a positive real number, we have such that So if we let , then and are both less than , so which completes the proof.