f be a bounded holomorphic function in the strip

V = {z in C:-1<Re(z)<1}

assume the limit lim_(y goes to infinity){f(i*y)}=c exists.

Prove that for each r in (0,1),

the limit lim_(y goes to infinity){f(x+i*y)}=c exists uniformly for -r<x<r

Need help, thanks a lot.