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Suppose ##a_n → α## and ##b_n → β##. Write ##a_n = α + x_n## and ##b_n = β + y_n ## where ##x_n## and ##y_n## are both null sequences. Now

$$

a_n + b_n = \alpha + x_n + \beta+y_n

$$

$$

a_n + b_n -(\alpha+ \beta) = x_n + y_n

$$

We have shown ##a_n + b_n - (α+ β)## is the sum of two null sequences and therefore also null. Hence, $$a_n + b_n \to α + β.$$

Are we really allowed to rewrite a sequence to another one that converges to the same limit, i.e ##a_n = α + x_n##?