Sameed
I think your proof needs to make it clear which contour the

is referring to in each case. So, for example, in the first case I think you should write:

to indicate the closed contour

which encloses all the poles (as stated in the exercise), and then:

(where n is a function of k.)
with an explanation of what the

are (maybe with a diagram?).
Also an explanation of how travelling round the

for all

is equivalent to travelling round

would be very useful to other members of the forum when trying to understand your proof.