Apply English pedantry to GAsyncInitable's docs

• "asynchronous" was misspelled as "asyncronous" in various places;
• punctuation was missing;
• g_async_initable_new_async() had a stray "and";
• references to g_async_initable_new_finish() were missing a "the".

https://bugzilla.gnome.org/show_bug.cgi?id=602417

gio/gasyncinitable.c

@@ -34,14 +34,14 @@
  * @include: gio/gio.h
  * @see_also: #GInitable
  *
- * This is the asynchronous version of #GInitable, it behaves the same
+ * This is the asynchronous version of #GInitable; it behaves the same
  * in all ways except that initialization is asynchronous. For more details
  * see the descriptions on #GInitable.
  *
  * A class may implement both the #GInitable and #GAsyncInitable interfaces.
  *
  * Users of objects implementing this are not intended to use the interface
- * method directly, instead it will be used automatically in various ways.
+ * method directly; instead it will be used automatically in various ways.
  * For C applications you generally just call g_async_initable_new_async()
  * directly, or indirectly via a foo_thing_new_async() wrapper. This will call
  * g_async_initable_init_async() under the cover, calling back with %NULL and
@@ -119,21 +119,21 @@ g_async_initable_base_init (gpointer g_iface)
  * Implementations may also support cancellation. If @cancellable is not
  * %NULL, then initialization can be cancelled by triggering the cancellable
  * object from another thread. If the operation was cancelled, the error
- * %G_IO_ERROR_CANCELLED will be returned. If @cancellable is not %NULL and
+ * %G_IO_ERROR_CANCELLED will be returned. If @cancellable is not %NULL, and
- * the object doesn't support cancellable initialization the error
+ * the object doesn't support cancellable initialization, the error
  * %G_IO_ERROR_NOT_SUPPORTED will be returned.
  *
- * If this function is not called, or returns with an error then all
+ * If this function is not called, or returns with an error, then all
  * operations on the object should fail, generally returning the
  * error %G_IO_ERROR_NOT_INITIALIZED.
  *
- * Implementations of this method must be idempotent, i.e. multiple calls
+ * Implementations of this method must be idempotent: i.e. multiple calls
  * to this function with the same argument should return the same results.
- * Only the first call initializes the object, further calls return the result
+ * Only the first call initializes the object; further calls return the result
- * of the first call. This is so that its safe to implement the singleton
+ * of the first call. This is so that it's safe to implement the singleton
  * pattern in the GObject constructor function.
  *
- * For classes that also support the #GInitable interface the default
+ * For classes that also support the #GInitable interface, the default
  * implementation of this method will run the g_initable_init() function
  * in a thread, so if you want to support asynchronous initialization via
  * threads, just implement the #GAsyncInitable interface without overriding
@@ -245,15 +245,15 @@ g_async_initable_real_init_finish (GAsyncInitable  *initable,
  * @user_data: the data to pass to callback function
  * @first_property_name: the name of the first property, or %NULL if no
  *     properties
- * @...:  the value if the first property, followed by and other property
+ * @...:  the value of the first property, followed by other property
  *    value pairs, and ended by %NULL.
  *
  * Helper function for constructing #GAsyncInitiable object. This is
- * similar to g_object_new() but also initializes the object asyncronously.
+ * similar to g_object_new() but also initializes the object asynchronously.
  *
  * When the initialization is finished, @callback will be called. You can
- * then call g_async_initable_new_finish() to get new object and check for
+ * then call g_async_initable_new_finish() to get the new object and check
- * any errors.
+ * for any errors.
  *
  * Since: 2.22
  */
@@ -289,11 +289,11 @@ g_async_initable_new_async (GType                object_type,
  * @user_data: the data to pass to callback function
  *
  * Helper function for constructing #GAsyncInitiable object. This is
- * similar to g_object_newv() but also initializes the object asyncronously.
+ * similar to g_object_newv() but also initializes the object asynchronously.
  *
  * When the initialization is finished, @callback will be called. You can
- * then call g_async_initable_new_finish() to get new object and check for
+ * then call g_async_initable_new_finish() to get the new object and check
- * any errors.
+ * for any errors.
  *
  * Since: 2.22
  */
@@ -332,11 +332,11 @@ g_async_initable_newv_async (GType                object_type,
  *
  * Helper function for constructing #GAsyncInitiable object. This is
  * similar to g_object_new_valist() but also initializes the object
- * asyncronously.
+ * asynchronously.
  *
  * When the initialization is finished, @callback will be called. You can
- * then call g_async_initable_new_finish() to get new object and check for
+ * then call g_async_initable_new_finish() to get the new object and check
- * any errors.
+ * for any errors.
  *
  * Since: 2.22
  */