Let Zn = (Xn, Yn) R for all n = {1, 2, 3,...}. Carefully explain...

 

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Let Zn = (Xn, Yn) € R² for all n = {1, 2, 3,...}. Carefully explain what [5] it means for a sequence {zn} to converge to a limit z E R². n=1 Consider the set U = {(x, y) = R2 : 2x + y< 1, x ≥ 0, y ≥ 0} and let [5] Zn = (Xn. Yn), where Xn= n and yn= n 1+2n' 1+ 4n Draw the set U in a graph. Write down the first three elements of the sequence {zn}_1 and identify these points in your graph. Show that xn <and yn <. Hence, and using your answer to part [10] (a), prove that {zn}=1 converges. Make sure you clearly state any additional results you use. Identify z = lim z, in your graph. Prove that Z, EU for all n. Hence, and using your answer to part [10] (c), prove that U is not closed. Is U open? Let Zn = (Xn, Yn) € R² for all n = {1, 2, 3,...}. Carefully explain what [5] it means for a sequence {zn} to converge to a limit z E R². n=1 Consider the set U = {(x, y) = R2 : 2x + y< 1, x ≥ 0, y ≥ 0} and let [5] Zn = (Xn. Yn), where Xn= n and yn= n 1+2n' 1+ 4n Draw the set U in a graph. Write down the first three elements of the sequence {zn}_1 and identify these points in your graph. Show that xn <and yn <. Hence, and using your answer to part [10] (a), prove that {zn}=1 converges. Make sure you clearly state any additional results you use. Identify z = lim z, in your graph. Prove that Z, EU for all n. Hence, and using your answer to part [10] (c), prove that U is not closed. Is U open?