Question

(1) Give a careful, detailed proof of the following Proposition. The sequence {2jnEN s unbounded Your proof should use the Archimedean Property and Russell's Paradox (2) Working directly from the basic definition of convergence to a ->0o Vn y together limit, show that limn-+ n- r and lim, imply that limn→х (2xn-3y.) 2x-3y (3) Give a proof, by induction, of the following Proposition. For 0 〈 n E N. suppose that the functions fı, . . . , f,: R → R are continuous. Then s continuous.

Answer 1

Given collect from the data let us assume that the equations are we have collected from the data are Now we have to take the equation are for 0 < n elementof N, these functions f_1 ellipsis f_n: R rightarrow R are the continuous then f_n compositefunction f_n - 1 compositefunction ellipsis f_2 compositefunction f_1 it is continuous let n = 1 f_1 is the continuous and its true for n = 1 Now we have to take n = k f_k compositefunction f_k - 1 compositefunction ellipsis f_2 compositefunction f_1 is continuous (1) let n = k + 1 f_k + 10 f_k compositefunction f_k - 1 compositefunction ellipsis f_2 compositefunction f_1 (2) we have f_k compositefunction f_k - 1 compositefunction f_2 compositefunction f_1 = g let g is the continuous on function then f_k + 1 compositefunction g {therefore from (2)} we have take the both continuous function f_k + 1 & g \r\n then from (1) is also continuous the