Let $(Tn)$ be a sequence of compact linear operators from a normed space $X$ into a Banach space $Y$. If $(Tn)$ is uniformly operator convergent i.e. $||T_n-T||$ tends to $0$, then show that the limit operator $T$ is compact.
I tried like this: Since $Y$ is Banach so $B(X,Y)$ is complete, and so $T$ is in $B(X,Y)$. So $T $ maps a bounded set $A$ to a bounded set $T(A)$. Now I want to prove
(a)for each sequence $(x_n)$ in $A$ , $T(x_n)$ has a Cauchy subsequence.
Or (b) $T(A)$ is relatively compact.
Kindly help (specially for (a)).