2. Let (V, ) be an F inner product space. Suppose (rn)N and (yn)N are convergent...

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2. Let (V, <, >) be an F inner product space. Suppose (rn)N and (yn)N are convergent sequences in V to x and y, respectively. Show that: (a) (< xn; Yn >)N is convergent in F to < x, y >. (i.e lim <axn, Yn >=< x, y >=< lim In, lim yn >, n00 whenever lim an = x and lim yn =y). n00 (b) lim < axn, z >= a < x, z > for all zEV and for all a E F. n00 2. Let (V, <, >) be an F inner product space. Suppose (rn)N and (yn)N are convergent sequences in V to x and y, respectively. Show that: (a) (< xn; Yn >)N is convergent in F to < x, y >. (i.e lim <axn, Yn >=< x, y >=< lim In, lim yn >, n00 whenever lim an = x and lim yn =y). n00 (b) lim < axn, z >= a < x, z > for all zEV and for all a E F. n00