Documentation: Update `readlink.md` to use `LibFileSystem` api

Base/usr/share/man/man2/readlink.md

@@ -26,12 +26,12 @@ and sets `errno` to describe the error.
 ## Notes
 
 The underlying system call always returns the full size of the target path on
-success, not the number of bytes written. `Core::DeprecatedFile::read_link()` makes use
+success, not the number of bytes written. `FileSystem::read_link()` makes use
 of this to provide an alternative way to read links that doesn't require the
 caller to pick a buffer size and allocate a buffer straight up.
 
 Since it's essentially impossible to guess the right buffer size for reading
-links, it's strongly recommended that everything uses `Core::DeprecatedFile::read_link()`
+links, it's strongly recommended that everything uses `FileSystem::read_link()`
 instead.
 
 ## Examples
@@ -40,7 +40,7 @@ The following example demonstrates how one could implement an alternative
 version of `getpid(2)` which reads the calling process ID from ProcFS:
 
 ```c++
-#include <LibCore/DeprecatedFile.h>
+#include <LibFileSystem/FileSystem.h>
 #include <unistd.h>
 
 pid_t read_pid_using_readlink()
@@ -53,11 +53,9 @@ pid_t read_pid_using_readlink()
     return atoi(buffer);
 }
 
-pid_t read_pid_using_core_file()
+ErrorOr<pid_t> read_pid_using_core_file()
 {
-    auto target = Core::DeprecatedFile::read_link("/proc/self");
-    if (target.is_null())
-        return -1;
+    auto target = TRY(File::read_link("/proc/self"sv));
     auto pid = target.to_uint();
     ASSERT(pid.has_value());
     return pid.value();